There has been some answers already, but it still may be interesting to look at the original paper (Mather et al., 1999) used by the FAO to make the chart. The sources are listed in the page here.

The Figure 1 in the Maher et al. paper shows the inverse relationship (in France) between the forest area (expressed as percentage of land area) and the population. From 1000 to the early 1800s, the growth of population resulted in encroachment on the forest. This phenomenon was reversed dramatically between 1350 and 1450 when, indeed, the population decreased due to the Plague. However, forest encroachment had been trending down before the Plague. For Marc Bloch (1931), this was the result of a general concern, shared by villagers and lords alike, about the disapperance of the forest as pasture grounds - forests were used for pasturing pigs and other livestock -, hunting grounds, and sources of timber. French kings were aware that the forest was not a limitless resource and they became concerned with its long-term sustainability, centuries before the concept became fashionable. Numerous ordinances were edicted from 1346 (Philippe VI) to 1414 (Charles V) - and many more in the following centuries - to organize the management of the French forests. The Article 4 of the first one, the "Ordinance of Brunoy", which organized the public administration of the forests, stipulated that

the Masters of the Waters and Forests shall survey and visit all the forests and woods and shall make the sales that are to be made there, with a view to ensuring that the said forests and woods can be perpetually maintained in good condition.

This does not change the global narrative: the renewed growth of French forests in the 15th century was caused by the death of part of the population, who stopped clearing them. But a small part was also due to State efforts to control and limit encroachment to keep forests sustainable, and those efforts continued in the next centuries.

Sources

• Bloch, Marc. Les Caractères Originaux de l’histoire Rurale Française. Armand Colin, 1931. http://classiques.uqac.ca/classiques/bloch_marc/histoire_rurale_fr_t1/histoire_rurale_fr_t1.html.
• Decruzy, Isambert, and Jourdan, eds. Recueil général des anciennes lois françaises : depuis l’an 420 jusqu’à la révolution de 1789. Tome IV. Belin-Le-Prieur, Verdiere, 1822. https://books.google.com/books?id=zJsFAAAAQAAJ&pg=PA522.
• Lormant, François. ‘Présentation historique de la législation forestière française’. Revue de Droit Rural, no. 2012–403 (1 May 2012). https://hal.univ-lorraine.fr/hal-02172936.
• Mather, A. S., J. Fairbairn, and C. L. Needle. ‘The Course and Drivers of the Forest Transition: The Case of France’. Journal of Rural Studies 15, no. 1 (1 January 1999): 65–90. https://doi.org/10.1016/S0743-0167(98)00023-0.

I believe you’ll find this answer from u/Udzu interesting while you wait for a more direct answer.

Massive depopulation caused by the Black Death.

In the 12th & 13th centuries, population increases led to the clear cutting of a lot of forest for arable land. Then in the 14th, drought and famine preceded the outbreak of the plague, which led to farmland being abandoned. People moved to cities. Then the plague hit a weakened population and it took centuries to recover. The farmland after time reverted to forest.

See this book for more, especially the “further reading” for the chapter on the Black Death.