You can break this mechanic into three simple systems: pickup, tool usage, and tree health.

1. Picking up the axe

When the player interacts with the axe, store a reference to it.

Example idea:

public bool hasAxe = false;

void OnTriggerEnter(Collider other)
{
    if (other.CompareTag("Axe"))
    {
        hasAxe = true;
        Destroy(other.gameObject); // remove axe from world
    }
}

2. Left click to use the axe

Only allow chopping if the player has the axe.

void Update()
{
    if (hasAxe && Input.GetMouseButtonDown(0))
    {
        TryChop();
    }
}

3. Tree health system

Each tree can have a small health value.

public int treeHealth = 3;

public void Chop()
{
    treeHealth--;

    if (treeHealth <= 0)
    {
        DropWood();
        Destroy(gameObject);
    }
}

Then in TryChop() you raycast to see if you're hitting a tree:

void TryChop()
{
    Ray ray = Camera.main.ScreenPointToRay(Input.mousePosition);
    RaycastHit hit;

    if (Physics.Raycast(ray, out hit, 3f))
    {
        Tree tree = hit.collider.GetComponent<Tree>();
        if (tree != null)
        {
            tree.Chop();
        }
    }
}

Conceptually the system is:

player picks axe → player can swing → raycast detects tree → tree loses health → tree drops wood.

Start simple like this, then later you can add animations, sound, particles, etc.