  We consider invasion percolation on a rooted regular tree. For the infinite
cluster invaded from the root, we identify the scaling behavior of its
$r$-point function for any $r\geq2$ and of its volume both at a given height
and below a given height. We find that while the power laws of the scaling are
the same as for the incipient infinite cluster for ordinary percolation, the
scaling functions differ. Thus, somewhat surprisingly, the two clusters behave
differently; in fact, we prove that their laws are mutually singular. In
addition, we derive scaling estimates for simple random walk on the cluster
starting from the root. We show that the invasion percolation cluster is
stochastically dominated by the incipient infinite cluster. Far above the root,
the two clusters have the same law locally, but not globally. A key ingredient
in the proofs is an analysis of the forward maximal weights along the backbone
of the invasion percolation cluster. These weights decay toward the critical
value for ordinary percolation, but only slowly, and this slow decay causes the
scaling behavior to differ from that of the incipient infinite cluster.
