index: write out the tree cache extension

Keeping the cache around after read-tree is only one part of the
optimisation opportunities. In order to share the cache between program
instances, we need to write the TREE extension to the index.

Do so, taking the opportunity to rename 'entries' to 'entry_count' to
match the name given in the format description. The included test is
rather trivial, but works as a sanity check.