MSC Eléonore Herquelot

PHD SÉBASTIEN BONENFANT Alice Guéguen

BS Rosemary Dray-Spira
email: rosemary.dray-spira@inserm.fr.

Impact of Diabetes on Work Cessation Data from the GAZEL cohort study

OBJECTIVE-To measure the impact of diabetes on work cessation, i.e., on the risks of work disability, early retirement, and death while in the labor force.

RESEARCH DESIGN AND METHODS-We used data from the GAZEL prospective cohort of 20,625 employees of the French national gas and electricity company "EDF-GDF." We identified 506 employees with diabetes and randomly selected 2,530 nondiabetic employed control subjects matched for major socio-demographic and occupational characteristics. Using a multistate Cox model, we estimated hazard ratios (HR) comparing the risks of transition from employment to disability, retirement, and death over time between participants with versus without diabetes.

RESULTS-Employment rate decreased more rapidly in participants with diabetes (51.9 and 10.1% at 55 and 60 years, respectively) compared with nondiabetic participants (66.5 and 13.4%, respectively). Participants with diabetes had significantly increased risks of transition from employment to disability (HR 1.7 [95% CI 1.0-2.9]), retirement ), and death ]) compared with participants without diabetes. Between 35 and 60 years, each participant with diabetes lost an estimated mean time of 1.1 year in the workforce (95% CI 0.99-1.14) compared with a nondiabetic participant.

CONCLUSIONS-Our results provide evidence for a profound negative impact of diabetes on workforce participation in France. Social and economic consequences are major for patients, employers, and society, a burden that is likely to increase as diabetes becomes more and more common in the working-aged population.

W

orldwide, 285 million people aged 20-79 years are estimated to have diabetes in 2010, and the prevalence is expected to rise substantially in the coming decades [START_REF]Diabetes Atlas[END_REF]. Diabetes confers an increased risk of disability [START_REF] Gregg | Diabetes and physical disability among older U.S. adults[END_REF][START_REF] Sinclair | Impact of diabetes on physical function in older people[END_REF][START_REF] Volpato | Women's Health and Aging Study. Comorbidities and impairments explaining the association between diabetes and lower extremity disability: The Women's Health and Aging Study[END_REF]. Moreover, complications of diabetes constitute major causes of morbidity-ranging from visual impairment to amputation to coronary heart disease and premature mortality [START_REF]Diabetes Atlas[END_REF].

Diabetes most often appears in middleaged adults, and the majority of individuals diagnosed with diabetes are of working age [START_REF]Diabetes Atlas[END_REF]. Several studies have shown that employment rate of people with diabetes is substantially reduced compared with the nondiabetic population in the U.S., with major economic consequences [START_REF] Hogan | Economic costs of diabetes in the US in 2002[END_REF][START_REF] Mayfield | Work disability and diabetes[END_REF][START_REF] Ng | Productivity losses associated with diabetes in the US[END_REF][START_REF] Smith | Productivity and economic burden associated with diabetes[END_REF]. Understanding the processes leading to such a reduced employment rate in the growing population of people with diabetes may provide insights into strategies for limiting the burden of the disease for individuals and society, although prospective studies aimed at understanding these processes have remained limited. Data from the Health and Retirement Study in the U.S. suggested that among working-aged individuals, work cessation occurred more frequently in those with diabetes versus those without prevalent diabetes at baseline in 1992 (9) as a result of higher rates of disability, retirement, and premature death [START_REF] Vijan | The impact of diabetes on workforce participation: results from a national household sample[END_REF]. However, these analyses were restricted to the prevalent cases of diabetes that were followed during a limited amount of time, and they did not account for occupational characteristics before diabetes occurrence. Thus they do not provide an accurate estimate of the overall relative burden of diabetes on work cessation.

The current study aimed at quantifying the impact of diabetes on work cessation. More specifically, our objectives were 1) to prospectively measure diabetes-associated risks of work disability, early retirement, and premature death from the time of diabetes onset, accounting for differences in occupational characteristics before diabetes onset and 2) to estimate the amount of time lost in employment for individuals with diabetes compared with those without diabetes.

RESEARCH DESIGN AND METHODS

Study design

The GAZEL cohort was established in 1989 among employees of the French national electricity and gas company "EDF-GDF." In January 1989, all male employees, then aged 40-50 years, and all women employees, then aged 35-50 years, were invited to participate in the cohort. Of the 44,922 subjects (31,411 men and 13,511 women) asked to participate, 20,625 (45.9%) accepted: 15,011 men and 5,614 women. Higher participation was associated with male sex, being married, having children, holding a managerial status, and residing in some regions of France [START_REF] Goldberg | Cohort profile: the GAZEL Cohort Study[END_REF]. EDF-GDF employees hold civil-servant-like status that entails job security. Typically, they are hired when they are in their 20s and stay with the company until retirement. Since enrollment, they have been followed prospectively through a yearly self-administered questionnaire, including a large set of information on health-such as self-reported diabetesand its determinants. Less than 1% of the participants has been lost to follow-up since 1989 [START_REF] Goldberg | Cohort profile: the GAZEL Cohort Study[END_REF].

Study population

In this study, we included GAZEL participants free of diabetes at baseline in 1989 who were subsequently classified as patients with diabetes during follow-up since they were still employed and a matched sample of nondiabetic participants.

Participants were classified as patients with diabetes if they consistently self-reported diabetes in the yearly questionnaire.

Consistent report of diabetes was defined based on the combination of response rate to the yearly questionnaire and frequency of self-reports of diabetes in the completed questionnaires. Those reporting diabetes inconsistently across time (i.e., less than once every 3 years following the first occurrence of self-reported diabetes) were classified as nondiabetic participants. For participants classified as patients with diabetes, the year of diabetes onset was that of the first occurrence of self-reported diabetes if the questionnaire of the preceding year indicated the absence of diabetes. In the case of missing questionnaires between the last year with no diabetes reported and the first year with diabetes reported, the year of diabetes onset was randomly assigned between these 2 years.

Nondiabetic participants were matched to those with diabetes for major socio-demographic and occupational determinants of workforce participation, i.e., sex and characteristics at the time of hiring in EDF-GDF, including calendar year, age, and occupational grade. Nondiabetic referents also had to be employed at the time they were matched to participants with diabetes. Five nondiabetic participants were randomly matched to each participant with diabetes.

Variables of interest

We obtained information on occupational status throughout the career in EDF-GDF (including characteristics at the time of hiring, retirement date, pension for disability or longstanding illness, and sickness absences) as well as on mortality from the company administrative records. Because retirement pensions are paid by EDF-GDF, these records provide comprehensive and accurate information. Legal age of retirement for EDF-GDF employees is between 55 and 60 years, mainly depending on the type of job: manual/unskilled workers are allowed to retire earlier. Participants were considered in disability if they had been granted a pension for disability or longstanding illness or if they had sick leave of more than 365 days.

BMI at age 50 years was obtained using self-reported height and weight and categorized as normal or underweight (,25.0 kg/m 2 ), overweight (25.0-29.9 kg/m 2 ), or obese (30.0 kg/m 2 or more).

Statistical analysis

Analysis was based on data collected as of 31 December 2007 and restricted to the age range of 35-60 years. Participants with diabetes were accounted for in the nonexposed group until diabetes onset and then in the exposed group starting from the year of diabetes onset. Probabilities of disability, retirement, and death at each point of the age range were estimated separately for participants with and without diabetes using the Kaplan-Meier method. Using a multistate Cox model, we estimated hazard ratios (HR) comparing the risks of transition from employment to each of these categories of inactivity between participants with versus without diabetes. These HR automatically accounted for characteristics used as matching criteria. In addition, further adjustment for BMI at age 50 years was performed. The proportional hazard hypothesis was verified based on the correlation between Schoenfeld residuals and time. Using the predicted probabilities of being in each of the four states of the model over time, we then estimated the difference in the amount of time spent out of the workforce between 35 and 60 years according to diabetes status. Confidence interval of this difference was estimated using the bootstrap percentile method.

Statistical analyses were performed using SAS 9.1 and R software with the packages survival and mstate.

RESULTS

Sample characteristics

Among the 20,625 participants included in the GAZEL cohort, 20,318 reported that they were free of diabetes in the baseline self-questionnaire in 1989. Among them, 981 were subsequently classified as patients with diabetes during follow-up. Of these 981 participants with incident diabetes, 508 were still employed at the time of diabetes onset and were thus considered as the exposed group in our study. Two of them were excluded because they did not have nondiabetic counterparts with comparable characteristics with regard to matching criteria, and each of the 506 remaining was matched to five nondiabetic participants. Thus the final sample consisted of 3,036 participants: 506 with diabetes and 2,530 without diabetes. As of 31 December 2007, 3.5% of these 3,036 participants were dead, 89.6% were retired, 0.8% were disabled, and 6.1% were still employed.

Characteristics of the study population according to diabetes status are shown in Table 1. As in the entire cohort, participants were mostly men. Half of them had been hired in EDF-GDF in the years 1957-1966 as they were in their 20s (mean age 22.8 years). The large majority (over 80%) had been hired on a low occupational position, i.e., either as manual workers or clerks. Upward occupational mobility was frequent, and as a result, only a minority (29.8% of participants with diabetes and 24.3% of those without diabetes) were still holding a low occupational position at 35 years.

Among the 506 participants with diabetes, mean age at diabetes onset was 51.6 years (minimum: 39; maximum: 60 years). At age 50 years 190 participants had diabetes; this number was 427 at age 55 years and 506 at age 60 years. Almost half (45.6%) of participants with diabetes were overweight, and one third (33.4%) were obese at 50 years. This was the case only for 40.7 and 5.7% of nondiabetic participants, respectively.

Employment status of participants with and without diabetes over time All of the 3,036 participants were employed as they were aged 35 years, and the large majority (95.3% of those with diabetes and 96.4% of those without diabetes) were still working as they reached the age of 50 years. Subsequently, employment rate decreased more rapidly in participants with diabetes (51.9 and 10.1% at ages 55 and 60 years, respectively) compared with those without diabetes (66.5 and 13.4%, respectively). At the age of 55 years, a higher proportion of participants with versus without diabetes were retired (39.5 vs. 28.2%, respectively; P , 0.001) or had died (2.9 vs. 1.1%, respectively; P = 0.003). At the age of 60 years, a higher proportion of participants with versus without diabetes were work disabled (5.0 vs. 1.4%, respectively; P , 0.001).

Frequency of work disability, retirement, and death over time

Overall, between the ages of 35 and 60 years, work disability occurred in 15 employed participants with diabetes and 162 without diabetes; retirement occurred in 399 employed participants with diabetes and 2,222 without diabetes; death occurred in 13 employed participants with diabetes and 22 without diabetes (Table 2).

Figure 1 shows the probabilities of disability, retirement, and death by diabetes status at each point of the age range between 50 and 60 years-the period Between the ages of 35 and 60 years, we estimated that each participant with diabetes lost an average time of 1.1 years in the workforce (95% CI 0.99-1.14) compared with a nondiabetic participant, whose retirement accounted for 0.70 year of this loss (95% CI 0.62-0.78), death for 0.28 year (95% CI 0.20-0.35), and disability for 0.09 year (95% CI 0.02-0.15).

Sensitivity analysis

Our definition of diabetes, based on a pattern of consistent self-report, is likely to have excluded the milder cases of disease. To capture these less severe cases, analyses were repeated using an alternative definition of diabetes in which all 773 cases with any mention of diabetes in the yearly questionnaire were classified as having diabetes, regardless of the consistency of this self-report. Although the effect of diabetes on disability was unchanged (HR 1.6 [95% CI 1.1-2.4]), the effects on retirement (HR 1.3 [1.1-1.4]) and death (HR 2.3 [1.2-4.2]) were lower than the estimates reported in Table 2. However, these effects remained significant. Likewise, the estimated average time of lost employment among participants with diabetes was decreased CONCLUSIONS-Our results provide evidence for a profound negative impact of diabetes on workforce participation in France. Indeed, we found that, among employees of the GAZEL cohort, diabetes is associated with an overall 60-70% increase in the risks of work disability and early retirement and with a sevenfold increase in the risk of death while in the labor force. The effect of diabetes on disability is largely attributable to obesity. These substantially increased risks of work cessation associated with diabetes translate into an absolute decrease of over 1 year spent in employment between 35 and 60 years for employees with diabetes compared with those without diabetes. Retirement accounts for the major part (64%) of this difference. Strengths of this study, which lend weight to these conclusions, include prospective data from a large occupational cohort, constituted of a diverse population of white-and blue-collar employees from all regions of France, with long-term follow-up that is almost 100% complete. Available information on occupational status throughout the career in the company is particularly comprehensive and accurate since it has been obtained directly from the company administrative records. In addition to ensuring highly reliable measures of the outcomes of interest, such information allowed us to account for occupational characteristics early in an individual's career in our analyses and thus to provide estimates of the burden of diabetes independent of these major occupational determinants of workforce participation. Our study has several limitations. First, information on diabetes was selfreported. Although accuracy of diabetes self-report is high [START_REF] Okura | Agreement between self-report questionnaires and medical record data was substantial for diabetes, hypertension, myocardial infarction and stroke but not for heart failure[END_REF], such a definition excludes undiagnosed cases of the disease, which are incorrectly considered as free of diabetes. However, only 13% of diabetic adults aged 55-74 years are estimated to be undiagnosed in France [START_REF] Bonaldi | A first national prevalence estimate of diagnosed and undiagnosed diabetes in France in 18-to 74-year-old individuals: the French Nutrition and Health Survey 2006/2007[END_REF]. Moreover, to be considered as patients with diabetes, participants in our study had not only to self-report diabetes once during follow-up but also to consistently self-report the disease over time after the first report. This probably improved the identification of true cases of diabetes compared with a definition based on a unique occurrence of self-reported diabetes. Although our definition may have led to exclude some cases of diabetes episodically self-reported, internally consistent results are obtained using either definition, as shown in our sensitivity analysis. In addition, available data did not allow us to distinguish between type 1 and type 2 diabetes. However, because our analyses were restricted to incident cases of diabetes occurring after enrollment in the GAZEL cohort, i.e., after the age of 35 years, our results mostly pertain to type 2 diabetes.

There are several considerations suggesting that the magnitude of diabetes impact on workforce participation may have been underestimated in our study. First, EDF-GDF employees have a particularly high level of job security; moreover, some categories of workers (e.g., those self-employed and agricultural workers), whose occupational status may be particularly vulnerable to health problems, are not represented in the company [START_REF] Goldberg | Cohort profile: the GAZEL Cohort Study[END_REF]. Second, to provide valid estimates of the impact of diabetes on work cessation, our analyses were restricted to incident cases of diabetes appearing after enrollment in the cohort, i.e., later than 1989. Thus cases of diabetes that occurred earlier in employees' careers were excluded. The impact of these early-onset cases on work cessation before 1989 could not be accounted for in our study since the GAZEL cohort was restricted to employees still working in 1989. However, a complementary analysis suggested that the diabetes-related increase in the risk of disability during follow-up may be more marked for cases prevalent in 1989 (results not shown). Third, because a large proportion of EDF-GDF employees are given the opportunity to retire between 55 and 60 years, retirement before 55 years rather than 60 years may constitute a better indicator of early retirement in this population. Diabetes-related increase in the risk of early retirement was substantially higher between 35 and 55 years (HR 2.1 [95% CI 1.7-2.5]) than between 35 and 60 years (HR 1.6 [95% CI 1.5-1.8]).

To our knowledge, this study is the first to provide evidence for a significant impact of diabetes on work cessation in France, a country with a system of universal social protection. Our findings of increased risks of disability, retirement, and death among employees with versus without diabetes are consistent with reports based on data from the Health and Retirement Study in the U.S. [START_REF] Tunceli | The impact of diabetes on employment and work productivity[END_REF][START_REF] Vijan | The impact of diabetes on workforce participation: results from a national household sample[END_REF]. This shows that such deleterious consequences of diabetes on workforce participation occur in the context of very different systems of social protection.

Diabetes is unevenly distributed across socioeconomic strata, with higher prevalence of the disease reported in the most deprived groups of the population [START_REF] Cowie | Sociodemographic characteristics of persons with diabetes[END_REF][START_REF] Espelt | Socioeconomic inequalities in diabetes mellitus across Europe at the beginning of the 21st century[END_REF]. This suggests that individuals with diabetes may be more prone than others to hold adverse occupational conditions, exposing them to an increased risk of unemployment. This could explain, at least in part, the differences formerly reported between individuals with versus without diabetes with regard to employment status [START_REF] Mayfield | Work disability and diabetes[END_REF][START_REF] Ng | Productivity losses associated with diabetes in the US[END_REF][START_REF] Smith | Productivity and economic burden associated with diabetes[END_REF][START_REF] Tunceli | The impact of diabetes on employment and work productivity[END_REF][START_REF] Vijan | The impact of diabetes on workforce participation: results from a national household sample[END_REF][START_REF] Kraut | Impact of diabetes on employment and income in Manitoba, Canada[END_REF]. In our study, all participants were employed by the same company and thus benefited from a comparable level of job security; moreover, we were able to provide estimates accounting for occupational characteristics before diabetes onset. Thus our findings add significant knowledge to the existing literature by providing evidence for an independent effect of diabetes on work cessation.

Further research is needed to ascertain the generalizability of our results outside the GAZEL cohort study, and more generally outside France. The French pattern of early transitions out of employment is characterized by a low age of retirement and by the importance of transitions through unemployment insurance and preretirement schemes, resulting in lower recourse to disability compared with other countries [START_REF] Salem | Labour force participation by the elderly and employment of the young: the case of France[END_REF]. Despite these differences, data from SHARE (Survey on Health Ageing and Retirement in Europe) suggest that the overall burden of health impairment on work cessation does not differ in France compared with other European countries [START_REF] Barnay | Effects of health on the labour force participation of older persons in Europe[END_REF]. This suggests that although the patterns of transitions out of employment are likely to be country specific, the negative impact of diabetes on workforce participation that we show might be fairly generalizable to other settings. Our results pertain to the population of working age, i.e., aged less than 60 years in France, a country with universal access to a high quality of medical care. The impact of diabetes on work cessation might be even greater in countries with older retirement ages or lower access to quality healthcare.

Diabetes can affect individuals' ability to maintain employment through different pathways. This can happen directly through impairment of bodily functions arising from diabetes complications, including loss of vision, amputations, or mobility limitations [START_REF] Kraut | Impact of diabetes on employment and income in Manitoba, Canada[END_REF][START_REF] Korff | Work disability among individuals with diabetes[END_REF]; workplace discrimination encountered by employees with diabetes may also play a role [START_REF] Mcmahon | Workplace discrimination and diabetes: the EEOC Americans with Disabilities Act research project[END_REF]. Moreover, because diabetes is frequently associated with other health problems, comorbidities can constitute additional barriers to employment. Obesity has been shown to impair employment status in several studies [START_REF] Paraponaris | Obesity, weight status and employability: empirical evidence from a French national survey[END_REF][START_REF] Renna | Direct and indirect effects of obesity on U.S. labor market outcomes of older working age adults[END_REF][START_REF] Rodbard | Impact of obesity on work productivity and role disability in individuals with and at risk for diabetes mellitus[END_REF][START_REF] Tunceli | Long-term effects of obesity on employment and work limitations among U.S. Adults, 1986 to 1999[END_REF]; in addition, our results suggest that differences in BMI between employees with versus without diabetes explain a significant part of the effect of diabetes on disability. Studies also suggest that macrovascular comorbid conditions [START_REF] Fu | Health care and productivity costs associated with diabetic patients with macrovascular comorbid conditions[END_REF] and depression [START_REF] Korff | Work disability among individuals with diabetes[END_REF] are associated with higher rates of unemployment among people with diabetes.

In conclusion, the current study provides evidence that diabetes substantially weighs on employees' chances of maintaining in employment. This effect of diabetes has major social and economic consequences for patients, employers, and society, a burden that is likely to increase as diabetes becomes more and more common in the working-aged population. Limiting the social and economic burden of diabetes is a major challenge to be addressed at different levels, including the health care system, employers, and social workers.

Figure 1 -

 1 Figure 1-Probability of work disability (A), retirement (B), and death (C) according to ages between 50 and 60 years among participants with (solid line) and without (dashed line) diabetes.

  Acknowledgments-No potential conflicts of interest relevant to this article were reported. E.H. ran all of the analyses and wrote the manuscript. A.G. provided statistical advice and reviewed and edited the manuscript. S.B. researched data and reviewed and edited the manuscript. R.D.-S. generated the study hypothesis and wrote the manuscript. care.diabetesjournals.org DIABETES CARE 5

Table 1 -

 1 Characteristics of the study population according to diabetes status

	Participants	Participants
	with diabetes	without diabetes

Values are n (%).

Table 2 -

 2 Risk of work disability, retirement, and death among participants with diabetes compared with those without diabetes

		Participants with diabetes	Participants without diabetes	HR [95% CI]
		Number	Incidence rate per	Number	Incidence rate per	Accounting for	Additionally adjusted
		of events	1,000 person-years	of events	1,000 person-years	matching criteria*	for BMI at age 50 years
	Employment
	to Disability	15	7.9	162	2.7	1.7 [1.0-2.9]	1.4 [0.8-2.4]
	to Retirement	399	209.1	2,221	37.6	1.6 [1.5-1.8]	1.6 [1.5-1.8]
	to Death	13	6.8	22	0.4	7.3 [3.6-14.6]	8.0 [3.8-17.0]
	care.diabetesjournals.org					DIABETES CARE

*Matching criteria were sex and calendar year, age, and occupational grade at the time of hiring in EDF-GDF.

The authors thank members of the GAZEL cohort study who contributed data for this study. The authors also thank the GAZEL cohort study team responsible for overseeing data collection and EDF-GDF (Service des Etudes Médicales, Service Général de Médecine de Contrôle).


Yves Aubry
email: yves.aubry@univ-tln.fr

Jean-Christophe Godin

Olivier Togni
email: olivier.togni@u-bourgogne.fr

CHOOSABILITY OF A WEIGHTED PATH AND FREE-CHOOSABILITY OF A CYCLE

Keywords: Mathematics Subject Classification. 05C15, 05C38, 05C72 Coloring, Choosability, Free-choosability, Cycles

A graph G with a list of colors L(v) and weight w(v) for each vertex v is (L, w)-colorable if one can choose a subset of w(v) colors from L(v) for each vertex v, such that adjacent vertices receive disjoint color sets. In this paper, we give necessary and sufficient conditions for a weighted path to be (L, w)-colorable for some list assignments L. Furthermore, we solve the problem of the free-choosability of a cycle.

Introduction

The concept of choosability of a graph, also called list coloring, has been introduced by Vizing in [START_REF] Vizing | Coloring the vertices of a graph in prescribed colors (in Russian), Diskret. Analiz. No. 29, Metody Diskret[END_REF], and independently by Erdős, Rubin and Taylor in [START_REF] Erdős | Choosability in graphs[END_REF]. It contains of course the colorability as a particular case. Since its introduction, choosability has been extensively studied (see for example [START_REF] Alon | Choosability and fractional chromatic number[END_REF][START_REF] Borodin | List edge and list colourings of multigraph[END_REF][START_REF] Thomassen | The chromatic number of a graph of girth 5 on a fixed surface[END_REF][START_REF] Zs | Every 2-choosable graph is (2m,m)-choosable[END_REF][START_REF] Gravier | A Hajós-like theorem for list coloring[END_REF] and more recently [START_REF] Gutner | Some results on (a:b)-choosability[END_REF][START_REF] Havet | Choosability of the square of planar subcubic graphs with large girth[END_REF]). Even for the original (unweighted) version, the problem proves to be difficult, and is NP-complete for very restricted graph classes. Existing results for the weighted version mainly concern the case of constant weights (i.e. (a, b)-choosability), see [START_REF] Alon | Choosability and fractional chromatic number[END_REF][START_REF] Erdős | Choosability in graphs[END_REF][START_REF] Gutner | Some results on (a:b)-choosability[END_REF][START_REF] Zs | Every 2-choosable graph is (2m,m)-choosable[END_REF]. For the coloring problem of weighted graphs, quite a little bit more is known, see [START_REF] Mcdiarmid | Channel assignement and weighted coloring[END_REF][START_REF] Havet | Channel assignement and multicolouring of the induced subgraphs of the triangular lattice[END_REF][START_REF] Havet | Finding a five bicolouring of a triangle-free subgraph of the triangular lattice[END_REF][START_REF] Kchikech | Approximation algorithms for multicoloring powers of square and triangular meshes[END_REF].

This paper considers list colorings of weighted graphs by studying conditions on the list assignment for a weighted path to be choosable. Starting from the idea that in a path, the lists of colors of non consecutive vertices do not interfere, and following the work in [START_REF] Godin | Coloration et choisissabilité des graphes et applications[END_REF], we introduce here the notion of a waterfall list assignment of a weighted path. It is a list assignment such that any color is present only on one list or on two lists of consecutive vertices. We show that any list assignment (with some additional properties) can be transformed into a similar waterfall list assignment. Then, using the result of Cropper et al. [START_REF] Cropper | Extending the disjoint-representatives theorems of Hall, Halmos, and Vaughan to listmulticolorings of graphs[END_REF] about Hall's condition for list multicoloring, we prove a necessary and sufficient condition for a weighted path with a given waterfall list L to be (L, w)-colorable (Theorem 9) and use it to derive (L, w)-colorability results for some general lists assignments.

In 1996, Voigt considered the following problem: let G be a graph and L a list assignment and assume that an arbitrary vertex v ∈ V (G) is precolored by a color f ∈ L(v). Is it always possible to complete this precoloring to a proper list coloring ? This question leads to the concept of free-choosability introduced by Voigt in [START_REF] Voigt | Choosability of planar graphs[END_REF].

We investigate here the free-choosability of the first interesting case, namely the cycle. As an application of Theorem 9, we prove our second main result which gives a necessary and sufficient condition for a cycle to be (a, b)-free-choosable (Theorem 12). In order to get a concise statement, we introduce the free-choice ratio of a graph, in the same way that Alon, Tuza and Voigt in [START_REF] Alon | Choosability and fractional chromatic number[END_REF] introduced the choice ratio (which equals the so-called fractional chromatic number).

In addition to the results obtained in this paper, the study of waterfall lists may be of more general interest. For now on, the method is extended in [START_REF] Aubry | Extended core and choosability of a graph[END_REF] to be used in a reduction process, allowing to prove colorability results on triangle-free induced subgraphs of the triangular lattice.

We recall in Section 2 some definitions related to choosability and freechoosability and introduce the definitions of the similarity between two lists and of a waterfall list that are fundamental for this paper. In Section 3, we show how to transform a list into a similar waterfall list and present a necessary and sufficient condition for a weigthed path to be choosable. Theses result are used in Section 4 to obtain conditions for the (L, w)colorability of a weighted path and for the (a, b)-free-choosability of a cycle.

Definitions and Preliminaries

Let G = (V (G), E(G)) be a graph where V (G) is the set of vertices and E(G) is the set of edges, and let a, b, n and e be integers.

Let w be a weight function of G i.e. a map w : V (G) → N and let L be a list assignment of G i.e. a map L : V (G) → P(N). By abuse of language and to simplify, we will just call L a list. If A is a finite set, we denote by |A| the cardinal of A.

A weighted graph (G, w) is a graph G together with a weight function w of G.

Let us recall the definitions of an (L, w)-colorable graph and an (a, b)free-choosable graph which are essential in this paper. Definition 1. An (L, w)-coloring c of a graph G is a map that associate to each vertex v exactly w(v) colors from L(v) such that adjacent vertices receive disjoints color sets, i.e. for all v ∈ V (G):

c(v) ⊂ L(v), |c(v)| = w(v),
and for all vv

′ ∈ E(G): c(v) ∩ c(v ′ ) = ∅.
We say that G is (L, w)-colorable if there exists an (L, w)-coloring c of G.

Particular cases of (L, w)-colorability are of great interest. In order to introduce them, we define (L, b)-colorings and a-lists. We define now the similarity of two lists with respect to a weighted graph: Definition 4. Let (G, w) be a weighted graph. Two lists L and L ′ are said to be similar if this assertion is true:

An (L, b)-coloring c of G is an (L, w)-coloring of G such that for all v ∈ V (G), we have w(v) = b. A a-list L of G is a list of G such that for all v ∈ V (G), we have |L(v)| = a.
G is (L, w)-colorable ⇔ G is (L ′ , w)-colorable.
The path P n+1 of length n is the graph with vertex set

V = {v 0 , v 1 , . . . , v n } and edge set E = n-1 i=0 {v i v i+1 }. To simplify the notations, L(i) denotes L(v i ) and c(i) denotes c(v i ).
By analogy with the flow of water in waterfalls, we define a waterfall list as follows:

Definition 5. A waterfall list L of a path P n+1 of length n is a list L such that for all i, j ∈ {0, . . . , n} with |i -j| ≥ 2, we have L(i) ∩ L(j) = ∅.
Notice that another similar definition of a waterfall list is that any color is present only on one list or on two lists of consecutive vertices. Figure 1 shows a list L of the path P 5 (on the left), together with a similar waterfall list L c (on the right). Definition 6. For a weighted path (P n+1 , w),

•

A list L is good if |L(i)| ≥ w(i) + w(i + 1) for any i, 1 ≤ i ≤ n -1. • The amplitude A(i, j)(L) (or A(i, j)) of a list L is A(i, j)(L) = ∪ j k=i L(k). L() 0 1 2 3 4 - similar L c () 0 1 2 3 4
Fig. 1. Example of a list L which is similar to a waterfall list L c .

In [START_REF] Cropper | Extending the disjoint-representatives theorems of Hall, Halmos, and Vaughan to listmulticolorings of graphs[END_REF], Cropper et al. consider Philip Hall's theorem on systems of distinct representatives and its improvement by Halmos and Vaughan as statements about the existence of proper list colorings or list multicolorings of complete graphs. The necessary and sufficient condition in these theorems is generalized in the new setting as "Hall's condition" :

∀H ⊂ G, k∈C α(H, L, k) ≥ v∈V (H) w(v),
where C = v∈V (H) L(v) and α(H, L, k) is the independence number of the subgraph of H induced by the vertices containing k in their color list. Notice that H can restricted to be a connected induced subgraph of G.

It is easily seen that Hall's condition is necessary for a graph to be (L, w)colorable. Cropper et al. showed that the condition is also sufficient for some graphs, including paths: (e) a triangle with an edge added at two of its three vertices.

This result is very nice, however, it is often hard to compute the left part of Hall's condition, even for paths. Hence, for our study on choosability of weigthed paths, we find convenient to work with waterfall lists for which, as we will see in the next section, Hall's condition is very easy to check.

waterfall lists

We first show that any good list can be transformed into a similar waterfall list.

Proposition 8. For any good list L of P n+1 , there exists a similar waterfall list L c with |L c (i)| = |L(i)| for all i ∈ {0, . . . , n}.

Proof. We are going to transform a good list L of P n+1 into a waterfall list L c and we will prove that L c is similar with L.

First, remark that if a color x ∈ L(i -1) but x ∈ L(i) for some i with 1 ≤ i ≤ n -1, then for any j > i, one can change the color x by a new color y ∈ A(0, n)(L) in the list L(j), without changing the choosability of the list. With this remark in hand, we can assume that L is such that any color x appears on the lists of consecutive vertices i x , . . . , j x . Now, by permuting the colors if necessary, we can assume that if x < y then i x < i y or i x = i y and j x ≤ j y .

Repeat the following transformation: 1. Take the minimum color x for which j x ≥ i x + 2 i.e. the color x is present on at least three vertices i x , i x + 1, i x + 2, . . . , j x ; 2. Replace color x by a new color y in lists L(i x + 2), . . . , L(j x ); until the obtained list is a waterfall list (obviously, the number of iterations is always finite). Now, we show that this transformation preserves the choosability of the list. Let L ′ be the list obtained from the list L by the above transformation.

If c is an (L, w)-coloring of P n+1 then the coloring c ′ obtained from c by changing the color x by the color y in the color set c(k) of each vertex k ≥ i x + 2 (containing x) is an (L ′ , w)-coloring since y is a new color.

Conversely, if c ′ is an (L ′ , w)-coloring of P n+1 , we consider two cases:

Case 1: x ∈ c ′ (i x + 1) or y ∈ c ′ (i x + 2). In this case, the coloring c obtained from c ′ by changing the color y by the color x in the color set c ′ (k) of each vertex k ≥ i x + 2 (containing y) is an (L, w)-coloring.

Case 2: x ∈ c ′ (i x + 1) and y ∈ c ′ (i x + 2). We have to consider two subcases:

• Subcase 1:

L ′ (i x + 1) ⊂ (c ′ (i x ) ∪ c ′ (i x + 1) ∪ c ′ (i x + 2)). There exists z ∈ L ′ (i x + 1) \ (c ′ (i x ) ∪ c ′ (i x + 1) ∪ c ′ (i x + 2)
) and the coloring c obtained from c ′ by changing the color x by the color z in c ′ (i x + 1) and replacing the color y by the color x in the color set c ′ (k) of each vertex k ≥ i x + 2 (containing y) is an (L, w)-coloring.

• Subcase 2: L ′ (i x + 1) ⊂ (c ′ (i x ) ∪ c ′ (i x + 1) ∪ c ′ (i x + 2)). We have |L ′ (i x + 1)| = (c ′ (i x ) ∪ c ′ (i x + 1) ∪ c ′ (i x + 2) ∩ L ′ (i x + 1) .
As c ′ is an (L ′ , w)-coloring of P n+1 , we have

|L ′ (i x +1)| = |c ′ (i x +2)∩L ′ (i x +1)|+|c ′ (i x +1)∩L ′ (i x +1)|+ c ′ (i x )\c ′ (i x +2) ∩L ′ (i x +1) , |L ′ (i x +1)|-w(i x +1)-|c ′ (i x +2)∩L ′ (i x +1)| = c ′ (i x )\c ′ (i x +2) ∩L ′ (i x +1) .
Since y ∈ c ′ (i x + 2) and y / ∈ L ′ (i x + 1), we obtain that

|c ′ (i x + 2) ∩ L ′ (i x + 1)| ≤ w(i x + 2) -1, hence |L ′ (i x + 1)| -w(i x + 1) -w(i x + 2) + 1 ≤ c ′ (i x )\c ′ (i x + 2) ∩ L ′ (i x + 1) .
But, by hypothesis, L is a good list. Thus

|L(i x + 1)| = |L ′ (i x + 1)| ≥ w(i x + 1) + w(i x + 2) and 1 ≤ c ′ (i x )\c ′ (i x + 2) ∩ L ′ (i x + 1) .
Consequently, there exists z ∈ c ′ (i x )\c ′ (i x + 2) ∩ L ′ (i x + 1). The coloring c is then constructed from c ′ by changing the color x by the color z in c ′ (i x + 1), the color z by the color x in c ′ (i x ) and the color y by the color x in the set c ′ (k) of each vertex k ≥ i x + 2.

The following theorem, which is a corollary of Theorem 7, gives a necessary and sufficient condition for a weighted path to be (L c , w)-colorable where L c is a waterfall list.

Theorem 9. Let L c be a waterfall list of a weighted path (P n+1 , w). Then P n+1 is (L c , w)-colorable if and only if:

∀i, j ∈ {0, . . . , n}, | j k=i L c (k)| ≥ j k=i w(k).
Proof. "if" part: Recall that A(i, j) = ∪ j k=i L c (k). For i, j ∈ {0, . . . , n}, let P i,j be the subpath of P n+1 induced by the vertices i, . . . , j. By Theorem 7, it is sufficient to show that ∀i, j ∈ {0, . . . , n},

x∈A(i,j) α(P i,j , L c , x) ≥ j k=i w(k).
Since the list is a waterfall list, then for each color x ∈ A(i, j), α(P i,j , L c , x) = 1 and thus x∈A(i,j)

α(P i,j , L c , x) = |A(i, j)| = | j k=i L c (k)|. "only if" part: If c is a (L c , w)-coloring of P n+1 then ∀i, j ∈ {0, . . . , n} : j k=i L c (k) ⊃ j k=i c(k). Since L c is a waterfall list, it is easily seen that | j k=i c(k)| = j k=i w(k). Therefore, ∀i, j ∈ {0, . . . , n} : | j k=i L c (k)| ≥ j k=i w(k).

Choosability of a path and free-choosability of a cycle

Theorem 9 has the following corollary when the list is a good waterfall list and |L(n)| ≥ w(n).

Corollary 10. Let L c be a waterfall list of a weighted path (P n+1 , w) such that for any i,

1 ≤ i ≤ n -1, |L c (i)| ≥ w(i) + w(i + 1) and |L c (n)| ≥ w(n). Then P n+1 is (L c , w)-colorable if and only if ∀j ∈ {0, . . . , n}, | j k=0 L c (k)| ≥ j k=0 w(k).
Proof. Under the hypothesis, if P n+1 is (L c , w)-colorable, then Theorem 9 proves in particular the result.

Conversely, since L c is a waterfall list of P n+1 , we have:

∀i, j ∈ {1, . . . , n}, |A(i, j)| = |∪ j k=i L c (k)| ≥ |∪ j k=i k-i even L c (k)| = j k=i k-i even |L c (k)|.
Since L c is a good list of P n+1 (for simplicity, we set w(n + 1) = 0): ∀i, j ∈ {1, . . . , n},

j k=i k-i even |L c (k)| ≥ j k=i k-i even (w(k) + w(k + 1)) ≥ j k=i w(k),
then we obtain for all i, j ∈ {1, . . . , n}, |A(i, j)| ≥ j k=i w(k). Since for all j ∈ {0, . . . , n}, |A(0, j)| ≥ j k=0 w(k), Theorem 9 concludes the proof.

Another interesting corollary holds for lists L such that |L(0)| = |L(n)| = b, and for all i ∈ {1, . . . , n -1}, |L(i)| = a. The function Even is defined for any real x by: Even(x) is the smallest even integer p such that p ≥ x. The above result is a starting tool used in [START_REF] Aubry | Extended core and choosability of a graph[END_REF] to attack McDiarmid and Reed's conjecture claiming that every triangle free induced subgraph of the triangular lattice is (9, 4)-colorable (hence the values a = 9 and b = 4 are somehow "natural"). It is also used in the following to determine the freechoice-ratio of the cycle.

The cycle C n of length n is the graph with vertex set V = {v 0 , . . . , v n-1 } and edge set E = n-1 i=0 {v i v i+1(mod n) }.

Let F C h (x) be the set of graphs G which are (a, b)-free-choosable for all a, b such that a b ≥ x:

F C h (x) = {G | ∀ a b ≥ x, G is (a, b)-free-choosable}.
Moreover, we can define the free-choice ratio fchr(G) of a graph G by: fchr

(G) := inf{ a b | G is (a, b)-free-choosable}.
If ⌊x⌋ denotes the greatest integer less or equal to the real x, we can state:

Theorem 12. If C n is a cycle of length n, then C n ∈ F C h (2 + n 2 -1
).

Moreover, we have:

fchr(C n ) = 2 + n 2 -1
.

Proof. Let a, b be two integers such that a/b ≥ 2 + ⌊ n 2 ⌋ -1 . Let C n be a cycle of length n and L a a-list of C n . Without loss of generality, we can suppose that v 0 is the vertex chosen for the free-choosability and

L 0 ⊂ L(v 0 ) has b elements. It remains to construct an (L, b)-coloring c of C n such that c(v 0 ) = L 0 . Hence we have to construct an (L ′ , b)-coloring c of P n+1 such that L ′ (0) = L ′ (n) = L 0 and for all i ∈ {1, ..., n -1}, L ′ (i) = L(v i ).
C n ∈ F C h (2 + n 2 -1
).

Hence, we have that fchr(C n ) ≤ 2 + ⌊ n 2 ⌋ -1 . Moreover, let us prove that M = 2 + ⌊ n 2 ⌋ -1 is reached.

For n odd, Voigt has proved in [START_REF] Voigt | On list Colourings and Choosability of Graphs[END_REF] that the choice ratio chr(C n ) of a cycle of odd length n is exactly M . Hence fchr(C n ) ≥ chr(C n ) = M , and the result is proved.

For n even, let a, b be two integers such that a b < M . We construct a counterexample for the free-choosability: let L be the list of C n such that

L(i) =        {1, . . . , a} if i ∈ {0, 1} {1 + i-1
2 a, . . . , ( i-1 2 + 1)a} if i = n -1 is odd {b + 1 + i-2 2 a, . . . , b + ( i-2 2 + 1)a} if i is even and i = 0 {1, . . . , b, 1 + ( n-4 2 + 1)a, . . . , ( n-4 2 + 2)a -b} if i = n -1

If we choose c 0 = {1, . . . , b} ⊂ L(0), we can check that it does not exist an (L, b)-coloring of C n such that c(0) = c 0 , so we could not do better.

Remark 13. In particular, the previous theorem implies that if n ≥ Even( 2b e ) then the cycle C n of length n is (2b + e, b)-free-choosable. Remark 14. Erdős, Rubin and Taylor have stated in [START_REF] Erdős | Choosability in graphs[END_REF] the following question: If G is (a, b)-colorable, and c d > a b , does it imply that G is (c, d)colorable ? Gutner and Tarsi have shown in [START_REF] Gutner | Some results on (a:b)-choosability[END_REF] that the answer is negative in general. If we consider the analogue question for free-choosability, then the previous theorem implies that it is true for the cycle.

Definition 2 .

 2 G is said to be (a, b)-choosable if for any a-list L of G, there exists an (L, b)-coloring c of G. Definition 3. G is said to be (a, b)-free-choosable if for any v 0 ∈ V (G), and for any list L of G such that for any v ∈ V (G) \ {v 0 }, we have |L(v)| = a and |L(v 0 )| = b, there exists an (L, b)-coloring c of G.

Theorem 7 (

 7 [START_REF] Cropper | Extending the disjoint-representatives theorems of Hall, Halmos, and Vaughan to listmulticolorings of graphs[END_REF]). For the following graphs, Hall's condition is sufficient to ensure an (L, w)-coloring:(a) cliques;(b) two cliques joined by a cut-vertex; (c) paths;(d) a triangle with a path of length two added at one of its vertices;

Corollary 11 . 2 otherwise

 112 Let L be a list of P n+1 such that |L(0)| = |L(n)| = b, and |L(i)| = a = 2b + e for all i ∈ {1, . . . , n -1} (with e = 0). If n ≥ Even 2b e then P n+1 is (L, b)-colorable.Proof. The hypothesis implies that L is a good list of P n+1 , hence by Proposition 8, there exists a waterfall list L c similar to L. So we get:∀i ∈ {1, . . . , n -1}, |L c (i)| ≥ 2b = w(i) + w(i + 1) and |L c (n)| ≥ b = w(n).By Corollary 10 it remains to prove that: ∀j ∈ {0, . . . , n}, |A(0, j)| ≥ j k=0 w(k) = (j + 1)b. Case 1: j = 0. By hypothesis, we have |A(0, 0)| = |L c (0)| ≥ b. Case 2: j ∈ {1, . . . , n -1}. Since L c is a waterfall list of P n+1 we obtain that: if j is even |A(0, j)| ≥ j k=0 k even |L c (k)| = b + j ∈ {0, . . . , n -1}, |A(0, j)| ≥ (j + 1)b.Case 3: j = n. Since n ≥ Even 2b e we deduce that |A(0, n)| ≥ (n + 1)b, which concludes the proof. For example, let P n+1 be the path of length n with a list L such that |L(0)| = |L(n)| = 4, and |L(i)| = 9 for all i ∈ {1, . . . , n -1}. Then the previous Corollary tells us that we can find an (L, 4)-coloring of P n+1 whenever n ≥ 8. In other words, if n ≥ 8, we can choose 4 colors on each vertex such that adjacent vertices receive disjoint colors. If |L(i)| = 11 for all i ∈ {1, . . . , n -1}, then P n+1 is (L, 4)-colorable whenever n ≥ 4.

  We have |L ′ (0)| = |L ′ (n)| = b and for all i ∈ {1, ..., n -1}, |L ′ (i)| = a. Since a/b ≥ 2 + ⌊ n 2 ⌋ -1 and e = a -2b, we get e/b ≥ ⌊ n 2 ⌋ -1 hence n ≥ Even(2b/e). Using Corollary 11, we get:


Jean Trap
email: trapjean@yahoo.fr

Fabrice Bureau

Alain Brethes

Bernard Jabiol

Jean-François Ponge

Matthieu Chauvat

Thibaud Decaëns

Michaël Aubert

Does moder development along a pure beech (Fagus sylvatica L.) chronosequence result from changes in litter production or in decomposition rates?

Keywords: Litter production, litter decomposition rate, humus form, nutrient availability, 130-yr chronosequence, Fagus sylvatica, loamy acidic soil, forest management

The development of temperate deciduous and conifers forests stands usually results in accumulation of forest floor organic matter and a shift from mull to moder humus forms. It has been suggested that an increase in nutrient uptake by trees during their rapid growth phase leads to topsoil acidification, decrease in earthworm density and thereby a decrease in litter turnover.

The focus question of the paper is whether the mull-moder shift occurring along forest ageing is due to higher leaf litter production and/or to lower decay rates of litter. The objectives of this research were to determine (1) changes in macro-morphological properties of humus forms, leaf litter production, litter decay rates, soil nutrients content and pH along a 130-year pure beech (Fagus sylvatica L.) chronosequence in Normandy (Northwest France), (2) if humus form varied from mull to moder with increasing stand age, and (3) if a shift from mull to moder resulted from increased litter production, decreased litter decay rates, or both. Annual litter production did not change significantly along the chronosequence (mean 2.41 t.ha -1 ). In contrast, litter decay rates decreased significantly during the rapid growth phase of trees. In consequence, the litter turnover time (1/k) was lower in the youngest stands (20 months) compared to the oldest ones (31 months). Even in the absence of a significant pattern of variation, litter production was positively correlated with the thickness of the OF (Oi) horizon. In contrast, litter decay was strongly negatively correlated with maximum thickness of the OH (Oa) horizon, suggesting that the appearance of the humification layer was mainly due to a decrease in litter decay rate. We did not find significant changes in the main properties of the organo-mineral horizon, suggesting that soil nutrient availability may not directly affect litter dynamics. We concluded that moder development along the chronosequence resulted in decreasing litter decay rates during the 4 aggradation phase while litter production was stable. Further studies are required to identify the ecological factors responsible for moder development along forest ageing.

Introduction

Successional patterns of humus forms have often been described along both temperate deciduous and conifers managed [START_REF] Aubert | Effect of tree mixture on the humic epipedon and vegetation diversity in managed beech forests (Normandy, France)[END_REF][START_REF] Chauvat | Humus structure during a spruce forest rotation: quantitative changes and relationship to soil biota[END_REF] and native forest maturation [START_REF] Bernier | Humus form dynamics during the silvigenetic cycle in a mountain spruce forest[END_REF][START_REF] Ponge | The heterogeneity of humus profiles and earthworm communities in a virgin beech forest[END_REF]. Early successional stages have been associated with fast organic matter turnover (translated into mull humus form) while lower organic matter (OM) recycling (expressed as moder occurrence) has been frequently observed under older forest on acidic soils (soil pH < 5.0) [START_REF] Ponge | Humus forms in terrestrial ecosystems: a framework to biodiversity[END_REF][START_REF] Salmon | Changes in humus forms and soil animal communities in two development phases of Norway spruce on an acidic substrate[END_REF]. More precisely, the appearance of moder with stands ageing consists in development of both fragmentation and humification litter layers, including high proportions of fine organic matter from fauna faeces.

Accumulation of organic materials on the forest floor results from unbalanced rates of two ecological processes: litter production and litter decomposition. For instance, the gradually higher amount of organic matter falling each year along stands ageing may not be fully incorporated within mineral horizons by soil organisms. Only soluble and non-lignified cell carbohydrates are totally utilized during the early phases of leaf litter decomposition [START_REF] Lavelle | Soil Ecology[END_REF][START_REF] Berg | Plant litter. Decomposition, humus formation, carbon sequestration[END_REF]. Each year, more and more recalcitrant compounds (lignified carbohydrates) may accumulate within the forest floor and promote the formation of fragmented and humified horizons [START_REF] Lebret | Litter production in an Atlantic beech (Fagus sylvatica L.) time sequence[END_REF].

Besides litter production, changes in litter decay rates may also lead to the appearance of a humification horizon. [START_REF] Ponge | Humus forms in terrestrial ecosystems: a framework to biodiversity[END_REF] and [START_REF] Ponge | The forest regeneration puzzle[END_REF] assumed that higher uptake of nutrients by trees during their phase of intense growth, for wood production, may lead to lower nutrient availability in the soil thereby affecting soil biological activity, especially earthworms activity. Indeed, there is also a global trend of decreasing abundance of earthworms during the phase of intense growth of temperate trees [START_REF] Bernier | Humus form dynamics during the silvigenetic cycle in a mountain spruce forest[END_REF][START_REF] Arpin | Diversity and dynamics of eco-units in the biological reserves of the Fontainebleau forest (France): Contribution of soil biology to a functional approach[END_REF]. This may lead to a decrease in litter decay rates promoting the accumulation of fragmentation and humification litter layers. Nevertheless, [START_REF] Hedde | Soil detritivore macroinvertebrate assemblages throughout a managed beech rotation[END_REF] did not find any significant changes in macrofaunal species turnover along a 200 years old pure beech chronosequence in France despite clear changes in humus forms suggesting that others factors than earthworm occurrence may be implicated.

The question of the paper was whether the shift mull-moder occurring along beech forest ageing is due to changes in leaf litter production and/or decay rates. More exactly, the objectives of this research were to determine (1) changes in macro-morphological properties of humus forms, leaf litter production, litter decay rates, soil nutrients content and pH along a 130-year pure beech (Fagus sylvatica L.) chronosequence Normandy (Northwest France), (2) if humus form changed from mull to moder with increasing stand age, (3) if a shift from mull moder resulted from increased litter production, decreased litter decay rates, or both. We also aimed at validating in a managed context the hypothesis developed by [START_REF] Ponge | Humus forms in terrestrial ecosystems: a framework to biodiversity[END_REF] about soil properties (nutrients content and pH) impact on moder appearance. We tested the hypothesis that fragmented and humified organic materials would accumulate within the forest floor due to decreasing litter decay rates and increasing litter production [START_REF] Lebret | Litter production in an Atlantic beech (Fagus sylvatica L.) time sequence[END_REF]. We also expected the decrease in nutrients availability and pH within the organo-mineral horizon with moder appearance [START_REF] Ponge | Humus forms in terrestrial ecosystems: a framework to biodiversity[END_REF][START_REF] Aubert | Effect of tree mixture on the humic epipedon and vegetation diversity in managed beech forests (Normandy, France)[END_REF].

Materials and methods

Study site

The study site was located in the Eawy state forest (France, Upper Normandy, 01°18' E; 49°44' N; 7200 ha). The climate is temperate oceanic with a mean annual temperature of +10°C and a mean annual precipitation of 800 mm. A space-for-time substitution procedure [START_REF] Pickett | Space-for-time substitution as an alternative to long-term studies[END_REF] was used to empirically reconstitute an even-aged forest chronosequence: sixteen pure beech stands were selected to represent four silvicultural phases (SP) [START_REF] Aubert | Diversity of plant assemblages in managed temperate forests: a case study in Normandy (France)[END_REF] of different ages: 13-18 years (SP15), 65-66 years (SP65), 91-103 years (SP95) and 121-135 years (SP130) (Table 1). Each phase was comprised of four replicated stands. All stands were managed as even-aged full-grown forest by the French Forestry Service (Office National des Forêts -ONF) and were located on a flat (plateau) topography at 205 m (a.s.l). The soil was an endogleyic dystric Luvisol [START_REF] Fao | World reference base for soil resources. A framework for international classification, correlation and communication[END_REF] developed on more than 80 cm of loess (lamellated silt of aeolian origin) lying on clay with flints [START_REF] Laignel | Variability of the clay fraction of the clay with flints of the western part of the Paris Basin[END_REF]. Understory vegetation was defined as a characteristic Endymio-Fagetum according to phytosociological classification [START_REF] Durin | Les hêtraies atlantiques et leur essaim climatique dans le nord-ouest et l'ouest de la France[END_REF]. At the centre of each stand, a 16 m² square plot was delimited away from vehicle tracks and base of trees to avoid any local acidification due to the influence of tree stems [START_REF] Beniamino | Soil acidification under the crown of oak trees .I. Spatial-distribution[END_REF].

Morphology of organic and organo-mineral horizons

Macro-morphological descriptions of organic and organo-mineral horizons were previously conducted within frames (25 cm x 25 cm) at three corners of the central plot according to the French nomenclature of soil horizons [START_REF] Jabiol | L'humus sous toutes ses formes[END_REF] in May 2007 (Table 2). We distinguished mull (mainly dysmull) and moder (hemimoder + eumoder + dysmoder) humus forms on the basis of morphological characters (Table 2). A total of 36 macro-morphological variables were described in the field on the basis of variation visible to the naked eye (Appendix 1) and 48 humus profiles were described (3 profiles per stand × 16 stands). We differentiated (1) the OL horizon consisting of almost unmodified leaf and woody fragments; (2) the OF horizon consisting of a mixture of coarse plant debris and fine organic matter (humus); (3) the OH horizon, which is an organic horizon characterized by an accumulation of decomposed plant litter and (4) the organo-mineral (A) horizon which vary in depth and structure among humus forms.

Chemical properties of the organo-mineral horizon

Four subsamples of the organo-mineral horizon were collected in May 2007 within frames (25 x 25 cm) located at each corner of the central square plot. In the laboratory, samples were sieved to 2 mm and air-dried until stable weight. Subsamples were dried at 105°C during 24h to determine water content. In air-dried samples of the organo-mineral horizon, concentrations of total C and N were measured by gas chromatography with a CHN pyrolysis micro-analyser (Flash 2000 Series, CHNS/O Analysers Thermo Scientific, IRCOF, France). Cto-N ratio, pH water and pH KCl (soil-to-solvent ratio = 1/2.5, WTW pH Meter 340, Weilheim, Germany) [START_REF] Baize | Guide des analyses en pédologie[END_REF], available P [START_REF] Duchaufour | Une méthode nouvelle de dosage du phosphore assimilable dans les sols forestiers[END_REF], and Cation Exchange Capacity (CEC) were also determined in the samples as well as total elements (Ca, Mg, K, Mn, Na, Al, Fe) by the cobalt hexamine exchange method [START_REF] Ciesielski | Determination of cation exchange capacity and exchangeable cations in soils by means of cobalt hexamine trichloride. Effects of experimental conditions[END_REF].

Litter production

Litterfall collectors, 1m² surface (1m side), 30cm deep and 1m height, were installed in order to characterize the annual litter production. Three collectors were placed in each stand near to the 16m² plot (less than 20m) along a 40m transect at 0, 20 and 40m. Litter was sampled every month from October 2007 to October 2008. A total of 48 collectors were installed (3 collectors per stand x 16 stands). Litterfall samples were oven-dried at 65°C for 48h [START_REF] Gardner | Water Content[END_REF]. Dry samples were sorted and litter components were classified into categories then weighed. Litter categories were leaves, wood fragments and reproductive organs. The wood category included dead wood and bark. Reproductive organs included beech mast and male flowers. Litter was stored at room temperature for litterbag experiment described below.

Litter decay

We sampled leaf litter in one stand of each silvicultural phase in November 2007 using a net (24m² surface) tight one meter above the forest floor in order to avoid any colonization by soil organisms. Litterfall samples were oven-dried at 65°C for 48h [START_REF] Gardner | Water Content[END_REF]) and stored at room temperature before the experiment. Litter decay was assessed in each silvicultural phase using the litterbag method [START_REF] Bocock | The disappearance of leaf litter under different woodland conditions[END_REF][START_REF] Verhoef | Litterbag method[END_REF]. Litterbags (15 x 20cm) were made from nylon net with 0.175 mm mesh size to only exclude macrofaunal activity [START_REF] Gartner | Decomposition dynamics in mixed-species leaf litter[END_REF][START_REF] Aubert | Aboveground-belowground relationships in temperate forests: plant litter composes and microbiota orchestrates[END_REF]. They were filled with 10g of dried leaves. This amount of litter corresponds to the mean annual leaf litter production in French beech forests according to [START_REF] Lebret | Litter production in an Atlantic beech (Fagus sylvatica L.) time sequence[END_REF]. Four different litterbag types were prepared, consisting of litter from four tree age-classes (15, 65, 95 and 130 years). In each stand, litterbags filled with the respective litter were inserted in the forest floor between OL and OF horizons. Litterbags were placed in the field in December 2007. For each silvicultural stage, 4 replicate bags (one per stand) were removed 12 months after the start of the experiment. When removed, litterbags were packed in plastic bags and transported to the laboratory for analyses. The content was then oven dried at 65°C for 48h to determine litter dry mass [START_REF] Gardner | Water Content[END_REF].

Data treatment and statistical analyses

The decay rate coefficient (k) estimates the disappearance of litter on an annual basis, using the following negative exponential decay function:

X t /X 0 =e -kt
where X 0 is the original mass of litter and X t is the mass remaining at time t [START_REF] Olson | Energy storage and the balance of producers and decomposers in ecological systems[END_REF]. The k value (month -1 ) was used to assess the turnover time of litter (1/k) [START_REF] Olson | Energy storage and the balance of producers and decomposers in ecological systems[END_REF].

All tests were computed with the R freeware (R Development Core Team, 2008) and statistical significance was set at P < 0.05. Means and standard deviations were calculated for each silvicultural phases (n=4 replicates). Comparisons of means were done among silvicultural phases using one-way ANOVA and Tukey HSD post-hoc tests. The normality of data and the homogeneity of variances were previously checked using Wilk-Shapiro [START_REF] Royston | An extension of Shapiro and Wilk's W test for normality to large samples[END_REF] and Bartlett tests (Bartlett, 1937), respectively. A Principal Component Analysis (PCA) was performed on organo-mineral horizon properties in order to present a clearer representation of topsoil parameters variations along the beech chronosequence (variables are listed in table 3). We also performed backward-stepwise multiple regressions to discern soil morphological variables that were significantly related to both litter production and litter decay along the chronosequence, after screening potential independent variables for significant co-variation (R²> 0.90).

Results

Litter production along the chronosequence

Due to high inherent variability, leaf production did not vary among silvicultural phases (Figure 1). It exhibited the lowest average value in SP15 (2.08 t.ha -1 ) and the maximal one in SP65 (2.58 t.ha -1 ). Similarly to leaf litter, wood production did not change between silvicultural phases. In contrast, the production of reproductive organs show significant variations with higher values in SP130 and lower ones in SP65. Leaf was the most abundant component of litterfall in all silvicultural phases. Wood was the second one except for SP130 where reproductive organs were most abundant (Figure 1). Total litter production (leaves + reproductive organs + wood) did not vary significantly among silvicultural phases (Figure 2) but exhibited its highest values in SP130 (2.82 t.ha -1 ) and its lowest ones in SP15 (2.27 t.ha -1 ).

Litter decomposition along the chronosequence

After 12 months of litterbag experiment, SP95 and SP130 exhibited a significantly higher remaining mass percentage (68 and 65%, respectively) compared to SP15 (53%) (Figure 2). The decay rate coefficient (k) was significantly higher in SP15 (0.050 month -1 ) compared to SP95 (0.032 month -1 ) while the turnover time (1/k) was significantly lower in SP15 (20 months) compared to older stands (31 and 28 months in SP95 and SP130, respectively) (Figure 3).

Chemical properties of the organo-mineral horizon

We did not observe any significant change in chemical properties in the organo-mineral horizon, except for the Mg pool (Table 3). Soil Mg content was higher in SP15 compared to SP65 and SP130. Even insignificant, CEC, Ca and Na contents as well as pH water and pH KCl decreased slightly along the chronosequence, while P, H and Al contents and ΔpH tended to increase.

The first two axes of the PCA performed on organo-mineral horizon data explained 61.1% of the total variation (Fig. 4.A). The first axis (relative inertia = 39.1%) was highly correlated with organic matter content, total C, total N, CEC and C/N ratio (positives scores). The second one (relative inertia = 21.9%) was highly correlated with pH, pH water , Al (negative scores) and Ca, Mg and Na (positive scores) (Fig. 4.A.). The individual factor maps (Figure 4.B) did not discriminate the silvicultural phases and humus forms.

Humus forms and litter dynamics co-variation patterns

Litter production was positively related to the percentage of bleached leaves in OLv, the maximum thickness of OF layer, the abundance of live roots and negatively related to the number of earthworm casts in OLn (P < 0.01, R² = 0.89) (Table 4.). In contrast, the litter decay rate (k) was negatively related to the maximum thickness of OF and OH horizons, the percentage of skeletonized and brown leaves in OLv (P < 0.05, R² = 0.62) (Table 4).

Discussion

Litter dynamics along the beech chronosequence

Our results showed that litterfall production did not change significantly along the pure beech chronosequence. The absence of significant differences agrees with [START_REF] Starr | Models of litterfall production for Scots pine (Pinus sylvestris L.) in Finland using stand, site and climate factors[END_REF] who did not find any correlation between stand age and litterfall production in Scots pine forest in Finland. Nevertheless, even insignificant, litterfall production tended to increase during the rapid growth phase of trees. This trend was also observed by [START_REF] Lebret | Litter production in an Atlantic beech (Fagus sylvatica L.) time sequence[END_REF]. In their study, beech litter production varied from 1.15 t ha -1 yr -1 in a thicket stand (10-yr-old) to 3.13 t ha -1 yr -1 in a mature stand (147-yr-old) from the Fougères forest in western France.

Besides climate and soil conditions, changes in litter production were often related to stand structure such as relative basal area, breast height diameter of trees or mean height and thinning intensity [START_REF] Ranger | The dynamics of biomass and nutrient accumulation in a Douglas-fir (Pseudotsuga menziesii Franco) stand studied using a chronosequence approach[END_REF][START_REF] Lebret | Litter production in an Atlantic beech (Fagus sylvatica L.) time sequence[END_REF][START_REF] Saarsalmi | Predicting annual canopy litterfall production for Norway spruce (Picea abies (L.) Karst.) stands[END_REF]. In the present study, we did not find any significant relationship between stand structure variables and litter production. Higher foliar biomass per tree in mature stands may compensate for decrease in tree density along the chronosequence, leading to constant litter production. However, we showed that the litter decomposition rate varied to a great extent along the chronosequence, the mean residence time of beech leaves passing from 20 months at 15 years to more than 30 months at 95 years, i.e. during the phase of rapid growth of beech trees. These changes in litter decay may be related to changes in the activity of soil organisms [START_REF] Berg | Decomposers: Soil Microorganisms and Animals[END_REF]. For instance, [START_REF] Ponge | Humus forms in terrestrial ecosystems: a framework to biodiversity[END_REF] explained that anecic earthworms populations are particularly sensitive to tree development due to soil acidification in the vicinity of tree trunks [START_REF] Ponge | The heterogeneity of humus profiles and earthworm communities in a virgin beech forest[END_REF][START_REF] Salmon | Changes in humus forms and soil animal communities in two development phases of Norway spruce on an acidic substrate[END_REF]. The abundance of anecic earthworms tends to decrease during the phase of intense growth of trees followed by progressive recovery as trees reach maturity then senesce [START_REF] Bernier | Humus form dynamics during the silvigenetic cycle in a mountain spruce forest[END_REF]. However, [START_REF] Hedde | Soil detritivore macroinvertebrate assemblages throughout a managed beech rotation[END_REF] did not find any significant macrofaunal species turnover along 200 years of pure beech chronosequence, using the same site as in the present study. The authors assessed that the lack of change in species composition along the chronosequence might reflect the confounding effect of silvicultural practices, i.e. monoculture of a soil-acidifying tree species on acidic soils and the use of tillage to assist natural regeneration. Those practices may have dramatic impacts on burrowing earthworms. The soil pH might be too much acid even in youngest stands (pH water 3.97) to permit high anecic earthworm abundance. Finally, according to [START_REF] Hedde | Soil detritivore macroinvertebrate assemblages throughout a managed beech rotation[END_REF] results, the observed changes in litter decay rates may not be related to earthworm community structure and activity.

Other explanation lies on the high lignin content of beech litter [START_REF] Karroum | Morphological evolution of beech litter (Fagus sylvatica L.) and biopolymer transformation (lignin, polysaccharides) in a mull and a moder, under temperate climate (Fougeres forest, Britany, France)[END_REF][START_REF] Sariyildiz | Variation in the chemical composition of green leaves and leaf litters from three deciduous tree species growing on different soil types[END_REF][START_REF] Hobbie | Tree species effects on decomposition and forest floor dynamics in a common garden[END_REF]. Fagus sylvatica is a woody species which produces low quality leaf litter with a high content of lignin and a low content of nitrogen (Sariyildiz and Anderson, 2003b;[START_REF] Hobbie | Tree species effects on decomposition and forest floor dynamics in a common garden[END_REF] compared to early successional species such as birch (Betula pendula) or hornbeam (Carpinus betulus). [START_REF] Karroum | Importance et devenir des biopolymères (lignines et polysaccharides) dans les sols d'une chronoséquence de hêtraies (Fagus sylvatica), en forêt de Fougères (France)[END_REF]2005) showed that lignin degradation occurring in organic layers of beech forests consists in an increase in the vanillic acid/aldhehyde ratio. The production of phenolic monomers during lignin degradation probably leads to strong acidification in both organic and organo mineral horizons during forest development. Consequently, microbial activity may vary and thus may affect litter decay rates. An investigation of litter quality along the chronosequence may provide insights in our understanding in moder development with forest ageing. It would be also interesting to investigate the structure and function of soil microbial communities in both the organic and organo mineral layers to validate this hypothesis [START_REF] Idol | Changes in microbial nitrogen across a 100-year chronosequence of upland hardwood forests[END_REF].

Humus forms and litter dynamics co-variation patterns

Even if litter production changes were not significant at P < 0.05, we showed that they were positively correlated with the maximum thickness of the OF layer, suggesting a possible effect of litterfall increase on moder development. They were also highly correlated with the percentage of bleached leaves (discolored leaves due to fungal activity) in the OLv horizon and were negatively correlated with the number of earthworm casts in the OLn horizon. It is probable that a higher beech leaf litter production favored the fungal community, in particular white-rot fungi [START_REF] Karroum | Morphological evolution of beech litter (Fagus sylvatica L.) and biopolymer transformation (lignin, polysaccharides) in a mull and a moder, under temperate climate (Fougeres forest, Britany, France)[END_REF] due to a higher amount of phenolic recalcitrant compounds (lignified carbohydrates and polyphenol-protein complexes) in litter, which may accumulate within the forest floor along the chronosequence. This hypothesis is supported by [START_REF] Karroum | Morphological evolution of beech litter (Fagus sylvatica L.) and biopolymer transformation (lignin, polysaccharides) in a mull and a moder, under temperate climate (Fougeres forest, Britany, France)[END_REF] who investigated the morphological evolution of beech litter and biopolymer transformation in humus forms from the Fougères forest (northwestern France). They showed that fungal attacks occurred mainly within the OL and OF horizons while bacterial activity was high within the OF and especially the OH horizons where they were responsible for the structural degradation of polysaccharides. The gradually higher amount with stands ageing of lignin-rich leaf litter falling each year may promote the fungal community within the OL layer, a spatial segregation of soil decomposers and consequently the thickening of the fragmentation layer in the absence of anecic earthworm activity [START_REF] Lebret | Litter production in an Atlantic beech (Fagus sylvatica L.) time sequence[END_REF][START_REF] Salmon | Changes in humus forms and soil animal communities in two development phases of Norway spruce on an acidic substrate[END_REF]. Our results also showed that litter production was significantly correlated with the abundance of live roots in the OLv horizon. We indeed observed a higher root colonization in OF and OH horizons along the studied chronosequence. This statement corroborated results by Ehrenfled et al. (1992) who found that as forest floor material accumulates along coniferous forest development, the percentage of fine roots increases within organic horizons. Since beech is a strongly ectomycorrhizal species [START_REF] Taylor | Fungal diversity in ectomycorrhizal communities of Norway spruce [Picea abies (L.) Karst.] and beech (Fagus sylvatica L.) along north-south transects in Europe[END_REF], the presence of roots in OF and OH horizons may promote the ectomycorhizal fungal community in organic litter layers and in turn limits decomposer activity [START_REF] Ehrenfeld | Live and dead roots in forest soil horizons: contrasting effects on nitrogen dynamics[END_REF].

We showed that litter decay rates were highly correlated with the maximum thickness of OH horizons, suggesting that the development of moder may result from the decrease in litter decay rates observed in SP95. The litter decay rate was also negatively correlated with the maximum thickness of the OF horizon and the percentage of skeletonized and brown leaves in the OLv horizon, suggesting that changes in mesofaunal and enchytraeid communities may take place along the chronosequence and greatly affect litter decay rates [START_REF] Chauvat | Humus structure during a spruce forest rotation: quantitative changes and relationship to soil biota[END_REF].

Soil nutrient availability as a controlling factor of humus form development?

We did not observe significant changes in nutrient availability within the organo-mineral horizon along the chronosequence. Furthermore, the PCA did not discriminate the four silvicultural phases and the humus forms. The first principal component described changes in organic matter content and CEC while the second one reflected changes in soil acidity. These results suggest that the variability in organo -mineral horizon properties among stands was not related to forest ageing. However, even insignificant, we observed a trend of decreasing soil quality (pH and nutrient availability) along the chronosequence, i.e. pH water was maximum in SP15 and minimum in SP95, pH KCl showed the same pattern, with 3.33 in SP15 and 3.09 in SP95, the Ca content in the organo mineral horizon was maximum in SP15 (1.43 cmol+ kg -1 ) and minimum in SP95 (0.71 cmol+ kg -1 ) while the Al content showed an inverse pattern. This trend agrees with results by [START_REF] Aubert | Effect of tree mixture on the humic epipedon and vegetation diversity in managed beech forests (Normandy, France)[END_REF] and [START_REF] Chauvat | How do soil fauna and soil microbiota respond to beech forest growth[END_REF]. They both observed an acidification of the organo mineral horizon along chronosequences of pure beech stands. The decrease in soil nutrient availability along forest maturation was often related to higher nutrient uptake by trees during the phase of intense growth of trees in response to nutrient requirements for wood building [START_REF] Brais | Changes in nutrient availability and forest floor characteristics in relation to stand age and forest composition in the southern part of the boreal forest of northwestern Quebec[END_REF][START_REF] Ponge | Humus forms in terrestrial ecosystems: a framework to biodiversity[END_REF]. Since mineral weathering (a stable process in mineral element production along forest maturation) may not supply enough nutrients (base cations) for beech tree requirements, soil quality might decrease along a pure beech chronosequence [START_REF] Brais | Changes in nutrient availability and forest floor characteristics in relation to stand age and forest composition in the southern part of the boreal forest of northwestern Quebec[END_REF][START_REF] Trap | Changes in soil N mineralization and nitrification pathways along a mixed forest chronosequence[END_REF], affecting in turn litter decay. This hypothesis is supported by Sariyildiz and Anderson (2003a) who showed that after 12 months of laboratory incubation, both oak (Quercus robur L.) and beech litter decomposition rates varied greatly according to soil quality. It can be wondered whether these changes in soil nutrient availability were significant enough to be responsible for the accumulation of organic material along the pure beech chronosequence.

Conclusions

Our results showed that the appearance of moder under the oldest stands was likely to be due to decreasing litter decay rates during the rapid growth phase of trees while litter production was constant. The first hypothesis formulated in the introduction is thus partially validated.

However, we cannot validate the second hypothesis concerning topsoil nutrients concentration and pH which should decrease along the pure beech chronosequence and limits soil biological activity (direct effect) and consequently litter decomposition rates. Indeed, we did not find any significant linear correlations between litter dynamics and organo-mineral horizon properties, and the PCA performed on those latter did not discriminate the silvicultural phases and humus forms.

Finally, we can conclude that soil nutrient availability effects on litter decay rates were either limited or not linear. These findings raise questions about the ecological factors responsible for the decrease of litter decay rates and thereby moder development with stand ageing which remain unknown. In the challenge to improve forest soil health and quality, further investigations are thus required, maybe by manipulating potential causative factors in long-term experiments. Vertical bars correspond to standard deviation. The decomposition constant k (month -1 ) was calculated from (X t /X 0 )=e -kt , where X 0 is the original amount of litter and X t is the amount of litter remaining at time t. Letters (a, b, x and y) refer to significant differences between silvicultural phases according to one-way ANOVA and Tukey HSD test (P < 0.05 level, n=4). 2004 2003 2003 2003 2003 2003 2003 2004 2003 2004 2002 2004 2002 2003 2004 2004 14.5 An ''X'' indicates that the variable was included in the model from the multiple regression analysis. The sign of the relationship between significant soil morphological variables and litter production or decay rates is indicated between brackets.

Figure captions

Appendix 1

SP15

Figure 1 .

 1 Figure1. Leaves, reproductive organs and wood production (expressed in t ha -1 yr -1 ) according to

Figure 2 .

 2 Figure2. Total litter production (t ha -1 yr -1 ) and final litter remaining mass (% of initial mass)

Figure 3 .

 3 Figure 3. Litter decomposition parameters according to silvicultural phases. Data are means.

Figure 4 .

 4 Figure 4. Principal Component Analysis (PCA) performed on organo-mineral horizon properties.

a

  Biomacrostructured; J : Juxtaposition A (A with massive or single-grain structure but no biological or chemical structure) : unmodified leaves less than one year old, OLv : unmodified leaves more than one year old, OF : coarse plant fragments with fine organic matter (FOM ), OH : more than 70% FOM , A: organic-mineral horizon. x : continuous; (xPercentage of basal area of beech (G/ha) compared to the total species present in the stand Last year cut Area (ha) Basal area (m²/ha) b With Du : Dysmull; H : Hemimoder; E : Eumoder; Do : Dysmoder (Jabiol et al. 2007) Vertical sequence c

  Figure1

Table 1 .

 1 Stand and soil characteristics of the stands in four silvicultural phases along the 130 year pure beech chronosequence in Eawy

	Forest, France.
	1	2	3	4	1	2	3	4	1	2	3	4	1	2	3	4
	15	20	15	15	67	67	67	68	105	93	103	93	137	137	123	123

Table 1

 1

Table 2 .

 2 Main properties of humus forms occurring along beech forest chronosequences according to[START_REF] Brêthes | Classification of forest humus forms: a french proposal[END_REF];[START_REF] Jabiol | L'humus sous toutes ses formes[END_REF];[START_REF] Gobat | Le sol vivant[END_REF] and correspondence with[START_REF] Green | Towards a taxonomic classification of humus forms[END_REF] classification.

	Properties	Humus forms
		Mull	Moder
		Dysmull	Hemimoder	Eumoder	Dysmoder
	OLn	×	×	×	×
	OLv	×	×	×	×
	OF	×	×	×	×
	OH			×/(×)	×
				<1cm	>1cm
	O/A transition	Discontinuity between O and A	O -A progressive transition
	A structure	Biomacrostructured	juxtaposition A $
	A pH	<5	<4.5	<4.5	<4.5
	A C/N ratio	15-20	20-25	20-25	≤25
	Soil fauna main groups	Anecic earthworms	Macro-	Mesofauna and insecta
		White rot fungi	arthropods	larvae
	Bacterial activity	+			-
	Fungal activity	-			+
	Green et al (1993)	Lepto-moder	Leptomoder / mormoder
	× : continuous horizon; (×) discontinuous horizon
	$ juxtaposition A : A with massive or single-grain structure but no biological or chemical structure;
	humified organic matter is juxtaposed to mineral matter

Table 2 Table 3 .

 23 Main properties of the organo mineral horizon (A) according to silvicultural phases in a pure beech chronosequence in Eawy forest, France.Data are means (SD). Letters (a and b) refer to significant differences among silvicultural phases according to Kruskal-Wallis rank sum test and Multiple comparison tests after Kruskal-Wallis test (P=0.05 level, n=4)

	Topsoil (A) properties	Silvicultural Phases
		SP15	SP65	SP95	SP130
	Total C (g kg -1 )	54.42 (24.92) a	41.81 (4.63) a	50.38 (14.80) a	45.92 (8.93) a
	Total N (g kg -1 )	3.26 (1.29) a	2.62 (0.28) a	3.04 (0.71) a	2.97 (0.52) a
	C/N (g kg -1 )	16.31 (1.16) a	15.92 (0.88) a	16.37 (1.46) a	15.29 (0.41) a
	P (g kg -1 )	0.17 (0.04) a	0.15 (0.05) a	0.21 (0.02) a	0.22 (0.17) a
	CEC (cmol+ kg -1 )	6.45 (0.97) a	5.40 (0.34) a	6.00 (0.73) a	5.73 (0.36) a
	H (cmol+ kg -1 )	0.90 (0.30) a	0.90 (0.05) a	1.14 (0.13) a	1.10 (0.15) a
	Ca (cmol+ kg 1 )	1.44 (1.05) a	0.54 (0.25) a	0.72 (0.39) a	0.39 (0.15) a
	Mg (cmol+ kg -1 )	0.37 (0.07) a	0.21 (0.03) b	0.27 (0.07) ab	0.22 (0.04) b
	Na (cmol+.kg -1 )	0.09 (0.05) a	0.06 (0.01) a	0.07 (0.01) a	0.07 (0.01) a
	K (cmol+ kg -1 )	0.21 (0.04) a	0.18 (0.03) a	0.26 (0.06) a	0.21 (0.05) a
	Fer (cmol+ kg -1 )	0.07 (0.06) a	0.07 (0.04) a	0.05 (0.02) a	0.08 (0.03) a
	Mn (cmol+ kg -1 )	0.19 (0.10) a	0.21 (0.10) a	0.23 (0.07) a	0.19 (0.05) a
	Al (cmol+ kg -1 )	3.79 (1.52) a	3.91 (0.49) a	4.14 (0.37) a	4.11 (0.32) a
	pH water	3.97 (0.38) a	3.97 (0.19) a	3.80 (0.06) a	3.86 (0.10) a
	pH KCl	3.33 (0.25) a	3.20 (0.06) a	3.09 (0.04) a	3.10 (0.09) a
	∆pH	0.64 (0.14) a	0.77 (0.14) a	0.71 (0.06) a	0.76 (0.12) a

Table 3 Table 4 .

 34 Stepwise multiple regressions with backwards elimination of litter production or decay rate with soil morphological variables.

	Soil morphological variables	Litter production	Litter decay rate (k )
	Leaf fragments in OLn	X(+)
	Earthworm casts in OLn	X(-)*
	Bleached leaves in OLv	X(+)**
	Live roots in OLv	X(+)*
	Skeletonized leaves in OLv		X(-)
	Brown leaves in OLv		X(-)
	OF maximum thickness	X(+)*	X(-)**
	OH maximum thickness		X(-)**
	P value	**	*
	R²	0.89	0.62

*** P < 0.001.

Acknowledgements

We are grateful to the "Conseil Régional de Haute-Normandie" for providing a PhD grant to J. Trap through the SCALE network of the GRR SER. This research was funded by both the SCALE network and the GIP ECOFOR (Viflorhum program). We want to thank P. Delporte, T. Tallec, B. Richard, G. Perez and P. Margerie (University of Rouen) for their assistance during litter collection and J.-F. Cheny (ONF) for his help in stand choice.

Appendix 1. Morphological variables used to determine humus forms within each stand of a 130yr-old pure beech chronosequence in Eawy forest, France. Living roots Aro From 0 to 3 (0: absent, 1: rare, 2: moderately present, 3: abundant)

a According to [START_REF] Jabiol | L'humus sous toutes ses formes[END_REF]. OLn: New. OLv: slightly altered.