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.