Dead trees have the exact same mechanics as regular trees, giving logs and always depleting after one section cut. The tree respawns between 36-60 seconds, randomly, and is not influenced by world population.^{[1]}
The chance to successfully chop is affected by the axe used and Woodcutting level, better axes and a higher woodcutting level increases the likelihood of successfully receiving a log. Kandarin headgear 1 and above gives double logs when cutting them.

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