STORY OF SEASONS: Pioneers of Olive Town

One of the tools provided by Victor during your first few days in Olive Town will be the axe. In upgrading this tool, it'll be easier to chop down trees of certain degrees. You will hardly come across many trees in the first area of your farmland, so you'll be better off repairing the bridge and getting into the later area(s). Or, you could just fell the trees that keep growing in the first area. Now, I do not know if this will count immature trees or stumps, but you'll be needing to fell tons of trees anyway for logs to upgrade later facilities, as well as levelling the skill itself. If you wish to keep track of this progress, you can check it in two different ways:

1) On the Title Screen, go through the trophy icon to find all tracked trophies. Trophies with cumulative progress can be tracked from here.
2) In-game, go to the Records page of your menu and choose Change Title. Interact with the Logging skill menu and you'll be able to see the number of trees you've chopped.