I'm not sure if this is an Emerald Tools issue and I'm not an expert in Minecraft mods, so sorry if I'm wrong.

I have Emerald Tools mod (EmeraldTools-Fabric-1.19.0-1.4.13), also I have Falling Tree mod (FallingTree-1.19-3.6.1) installed both together.
When I chop trees using stone axe from vanilla Minecraft, trees are falling as expected.
When I chop trees using Copper Axe from Emerald Tools, trees are not falling. I could of course add axes from Emerald Tools to allowed tools list in Falling Tree mod settings.

Falling Tree mod have hint in its settings: Tools with "axe" tag will be automatically added to allowed tools list
So I assumed that axes from Emerald Tools does not have axe tag properly set. Please check it and fix if required.

This also could be an issue of Falling Trees mod for sure.

The fabric "standard is "axes", tags like items should be plural.

My axes are already listed.

In fact, if they are not in the list, they won't work as axes in Fabric.

It looks like the Harvester code simply checks if it is an AxeItem, which mine are. Something else may be going on.

BTW I checked axes from More Gems mod (I assume it's also is your mod just like Emerald Tools), f.e. Sapphire Axe, and these axes work as expected with Falling Tree mod.

I fired up the mod and looked at its config, it automatically picked up all of my axes.

What does your file look like?

Didn't get what exactly file and in which mod should I check? Could you please specify exact mod name and file name which I should check? Thanks

in the folder where you minecraft instance is installed.

BTW: with my basic setup, all of the axes work fine at harvesting

OK, may be there is some misunderstanding:
I use mod FallingTree (see https://www.curseforge.com/minecraft/mc-mods/falling-tree) to chop whole trees, and you (I suppose) do mention Tree Harvester mod (https://www.curseforge.com/minecraft/mc-mods/tree-harvester-fabric).
They are different mods. I don't use Tree Harvester.

My bad. I will try FallingTree. According to the author it automatically picks up tools that extend AxeItem, which mine do.

I just tested my mods (EmeraldTools, MoreGems, Gobber) and all the axes work fine with FallingTree.

The tree axes work as well.

Thanks, may be it's an issue in my mod setup or mods combination.

I've tested it on a reduced amount of mods (only they was installed):

architectury-5.7.27-fabric.jar
cloth-config-7.0.72-fabric.jar
EmeraldTools-Fabric-1.19.0-1.4.13.jar
fabric-api-0.56.0+1.19.jar
FallingTree-1.19-3.6.0.jar
modmenu-4.0.0.jar
MoreGems-Fabric-1.19.0-1.4.19.jar
notenoughcrashes-4.1.6+1.19-fabric.jar
RoughlyEnoughItems-9.0.489.jar

and everything was fine.

It seems, some other mod breaks this behavior. Thanks.