asymptotic behavior of random walks on a group, such as decay of return probability and growth of entropy, reflects large scale geometry of the underlying group. the notion of isoperimetric profiles, motivated by classical isoperimetric inequalities and faber-krahn inequalities, captures useful geometric information that affects behavior of random walks. in this talk i will explain some interesting examples of groups acting on trees, where random walks exhibit rich exotic behavior. a connection between return probability lower bound and entropy growth will also be discussed.