As of Beta 7.2.3 trees will regrow naturally if you remove all the wood pulp and tree stumps.
Remember to leave at least some trees around in the area however because if you deforest an entire area they will not grow back.
Seedlings can be obtained by striking leaves on a tree with an axe before chopping the tree down.
The default tree seed drop chance is 30%. As of Beta 7.2.3, Only Cedar, Birch and Oak seeds can be acquired in this way.
In version 8.0, tree seeds are gathered by clearing the Tree Debris that remains after felling a tree.
Types of Trees
There are many types of trees in ECO: