How wood is produced
Forestry lodge building generates wood using surrounding living trees as a natural endless source.
Water and electricity are basic requirements, advanced forestry demands liquid fertilizer instead of water to greatly increase efficiency of all operations.
What trees make sense for production efficiency
The amount of trees (not dry or dead) in range of the building determines the efficiency of wood production recipe and pollution absorption. Range is shown with a circle.
Maximum managing capacity is 60 trees for basic tier and 130 trees for advanced lodge tier. Therefore upgrading the building to tier-2 can decrease the recipe efficiency for some time if living trees number was exactly 60 or lower.
How to increase forest density
Trees slowly grow when forestry is working (no matter how effectively) and seedlings are present in the output inventory. The growth speed depends on total amount of trees in operational range (dead&dry and associated with other lodges too). Fastest growth speed takes place when all surrounding trees number is equal to approx. 45 and 90 for basic and advanced lodge tier accordingly. Thus dead/dry trees are better to keep before reaching that value and cut after exceeding it. There must be at least one living tree associated with this building. New trees appear near existing ones.
Another approach is to grow saplings out of seedlings in the greenhouse, then plant them within the range of the working building. This way forest can be created from scratch and trees locations are selected by the player. Sapling needs additional space around it while growing. Tree won't appear on inappropriate ground.
Factors decreasing forestry efficiency
Multiple buildings with overlapping areas (unless the area is a very dense forest and there are enough trees for all of them) - each living tree can be managed by only one building. Keep the distance or build them near (or inside) natural dense forests.
Poisoned trees https://wiki.factorio.com/Tree#Pollution_removal. Heavy pollution (really heavy) slowly hurts trees thus progressing the recession of natural trees: they absorb less pollution, loose some leaves. Working forestry lodge cuts such trees (not immediately, but severely damaged trees are removed sooner), collecting wood and decreasing the trees amount in range. For some unknown reason at game start some number of trees in the starting location is already poisoned (may be it's caused by spaceship crash?).
Why liquid fertilizer is used
This is factorio engine limitation. Also it replaces water supply.
Should seedlings be taken or kept in inventory
It's better to always keep 10..20 seedlings in the output inventory. One inserter can take out only wood, another one only seedlings with a condition of exceeding items number. Hint for a fresh built forestry lodge: factorio2 allows to put items into output inventory.