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Recently, D. Vuki$\check{c}$evi$\acute{c}$ and J. Sedlar in \cite{Vuki}introduced an order "$\preceq$" on $\mathcal{T}_n$, the set of trees on $n$vertices, such that the topological index $F$ of a graph is a function definedon the order set $\langle\mathcal{T}_n,\preceq\rangle$. It provides a newapproach to determine the extremal graphs with respect to topological index$F$. By using the method they determined the common maximum and/or minimumgraphs of $\mathcal{T}_n$ with respect to topological indices of Wiener typeand anti-Wiener type. Motivated by their researches we further study the orderset $\langle\mathcal{T}_n,\preceq\rangle$ and give a criterion to determine itsorder, which enable us to get the common extremal graphs in four prescribedsubclasses of $\langle\mathcal{T}_n,\preceq\rangle$. All these extremal graphsare confirmed to be the common maximum and/or minimum graphs with respect tothe topological indices of Wiener type and anti-Wiener type. Additionally, wecalculate the exact values of Wiener index for the extremal graphs in the ordersets $\langle\mathcal{C}(n,k),\preceq\rangle$,$\langle\mathcal{T}_{n}(q),\preceq\rangle$ and$\langle\mathcal{T}_{n}^\Delta,\preceq\rangle$.

The extremal graphs of order trees and their topological indices