A surprising observation
A certain class of lambdaterms is "the same shape as" their types.
Specifically, the class of lambdaterms is BCIterms in normal form.
 A lambdaterms is a BCIterm if each lambdanode is the target of a leaf, and no lambdanode is the target of more than one leaf.

A lambdaterm is in normal form if the left child of an applynode is never a lambdanode.
If you know the definition of betareduction, it is easy to see that the set of BCIterms is closed under betareduction.