Know more

About cookies

What is a "cookie"?

A "cookie" is a piece of information, usually small and identified by a name, which may be sent to your browser by a website you are visiting. Your web browser will store it for a period of time, and send it back to the web server each time you log on again.

Different types of cookies are placed on the sites:

  • Cookies strictly necessary for the proper functioning of the site
  • Cookies deposited by third party sites to improve the interactivity of the site, to collect statistics

Learn more about cookies and how they work

The different types of cookies used on this site

Cookies strictly necessary for the site to function

These cookies allow the main services of the site to function optimally. You can technically block them using your browser settings but your experience on the site may be degraded.

Furthermore, you have the possibility of opposing the use of audience measurement tracers strictly necessary for the functioning and current administration of the website in the cookie management window accessible via the link located in the footer of the site.

Technical cookies

Name of the cookie


Shelf life

CAS and PHP session cookies

Login credentials, session security



Saving your cookie consent choices

12 months

Audience measurement cookies (AT Internet)

Name of the cookie


Shelf life


Trace the visitor's route in order to establish visit statistics.

13 months


Store the anonymous ID of the visitor who starts the first time he visits the site

13 months


Identify the numbers (unique identifiers of a site) seen by the visitor and store the visitor's identifiers.

13 months

About the AT Internet audience measurement tool :

AT Internet's audience measurement tool Analytics is deployed on this site in order to obtain information on visitors' navigation and to improve its use.

The French data protection authority (CNIL) has granted an exemption to AT Internet's Web Analytics cookie. This tool is thus exempt from the collection of the Internet user's consent with regard to the deposit of analytics cookies. However, you can refuse the deposit of these cookies via the cookie management panel.

Good to know:

  • The data collected are not cross-checked with other processing operations
  • The deposited cookie is only used to produce anonymous statistics
  • The cookie does not allow the user's navigation on other sites to be tracked.

Third party cookies to improve the interactivity of the site

This site relies on certain services provided by third parties which allow :

  • to offer interactive content;
  • improve usability and facilitate the sharing of content on social networks;
  • view videos and animated presentations directly on our website;
  • protect form entries from robots;
  • monitor the performance of the site.

These third parties will collect and use your browsing data for their own purposes.

How to accept or reject cookies

When you start browsing an eZpublish site, the appearance of the "cookies" banner allows you to accept or refuse all the cookies we use. This banner will be displayed as long as you have not made a choice, even if you are browsing on another page of the site.

You can change your choices at any time by clicking on the "Cookie Management" link.

You can manage these cookies in your browser. Here are the procedures to follow: Firefox; Chrome; Explorer; Safari; Opera

For more information about the cookies we use, you can contact INRAE's Data Protection Officer by email at or by post at :


24, chemin de Borde Rouge -Auzeville - CS52627 31326 Castanet Tolosan cedex - France

Last update: May 2021

Menu Logo Principal AgroParisTech Université de Lorraine UMR SILVA

Home page

Size-density trajectories for even-aged sessile oak (Quercus petraea (Matt.) Liebl.) and common beech (Fagus sylvatica L.) stands revealing similarities and differences in the mortality process

11 March 2020

Ningre, F. ; Ottorini, J. M. ; Le Goff, N.
Annals of Forest Science, 2019, 76 (3)
Pièces jointes : Ningre2019_Article_Size-densityTrajectoriesForEve.pdf
Key message We studied the size-density trajectories of pure even-aged unthinned experimental sessile oak ( Quercus petraea (Matt.) Liebl.) stands in the ranges of 994–135,555 trees per hectare initial densities, observed from the ages of 5 to 38. We compared them to unthinned beech ( Fagus sylvatica L.) stands from the same experimental area. An original piecewise polynomial function was fitted to the trajectories, giving way to various applications. For each species, the initial number of trees per hectare ( N 0 ) and the mean girth at breast height at the onset of mortality (Cg 0 ) were parameters of the trajectory model, in addition to the parameters of the maximum size-density lines. The two former parameters (Cg 0, N 0 ) were tied by a linear relationship, which allowed the prediction of trajectories for initial densities not included in the study data. For oak and beech, mortality onset occurred at a constant relative density (RDI), for all initial stand densities, respectively, 0.35 and 0.29. The comparison of the size-density trajectories of oak and beech allowed to establish that oak needs more space than beech for comparable mean girth, and then is less efficient than beech in its space requirements. Context This paper models the size-density trajectories of pure even-aged sessile oak stands, including the early development stage. It compares the oak results with those on common beech on the same site from a previous study. Aims A novel approach to size-density trajectories, with an original polynomial piecewise function previously used for beech stands on the same site, was satisfactorily used again as a mortality model to provide references to managers of oak forests. Material and methods A 38-year-old oak spacing trial, re-measured from year 5 to year 38, provided the opportunity to study the size-density trajectories of unthinned stands of this species. Results The fit of the piecewise polynomial function allowed us to estimate the parameters of the size-density trajectories of all stands, which were the initial number of trees per hectare (N0) and the mean girth at breast height at the onset of mortality (Cg0), in addition to the intercept (a) and slope (b) of the maximum size-density line. A linear relationship between Ln(N0) and Ln(Cg0) (where Ln is the Neperian logarithm) allowed us to reduce the number of parameters needed to fit the trajectories and made it possible to predict a size-density trajectory from any initial density not observed in the experimental stands. Moreover, this later line appeared to be parallel to the maximum size-density line, and new data allowed to establish that this was also the case for the beech stands on the same site. This parallelism feature translates to the onset of mortality occurring at the same relative density for stands of every initial density that is 0.35 for oak and 0.29 for beech. Conclusion Given the parameters of the maximum size-density line, a single-parameter function family could be used to predict the size-density trajectories of oak stands. The predicted trajectories have various applications in oak silviculture and growth simulators. The oak data and new data for beech stands on the same site allowed to compare the two species and draw conclusions on similitudes and differences concerning mortality and space requirements of both species.