Leaves build a Merkle tree
Each branch is hashed as a TapLeaf and combined into TapBranch hashes. The root tweaks the internal key. Tree shape affects the proof size for each branch, so likely paths can be placed at shallower depths.
The control block proves membership
A script-path witness provides stack data, the executed script and a control block containing the internal key, parity and sibling hashes. The verifier rebuilds the commitment and rejects a mismatch.
Tapscript has its own rules
BIP 342 uses Schnorr signatures and disables the old CHECKMULTISIG opcode for Tapscript. CHECKSIGADD makes threshold constructions possible by adding successful signature checks before comparing a total.
Future versions need careful handling
Leaf versions and unknown public keys reserve upgrade space. That flexibility is not permission for wallet software to improvise: policy tools must understand the executed version, resource limits and recovery branches before receiving funds.