Link-cut tree is a data structure invented by Sleator and Tarjan, which supports various operations and queries on a $n$-node forest in time $O(\log n)$. (For example, operation link combines two trees in the forest into one, while operation cut divide a tree in the forest into two trees.)
Several applications are known by using link-cut trees, and here I'm particularly interested in Goodrich's separator decomposition, which given a $n$-node plane graph $G$ one can obtain a corresponding binary tree where nodes are subgraphs of $G$, and the children of a node $H$ are the subgraphs of $H$ divided by the separator on $H$. Such a decomposition can be easily constructed in $O(n \log n)$ time (since a separator can be found in $O(n)$ time, and since the separator divides the graph so balanced, after $O(\log n)$ level of separations the leaves of the tree are of size $O(1)$). The main contribution of Goodrich is that he can construct such a decomposition in time $O(n)$, by maintaining and reusing the data structures used for finding separators in each level.
One of the data structures being used in the construction, is indeed the link-cut tree. In page 7 of the paper by Goodrich, he claimed that the initialization of the link-cut tree can be done in time $O(n)$. While I go through all the papers cited there, it seems to me that if we construct a link-cut tree via operation link, it takes time $O(n \log n)$ in total.
Do I misunderstand something? Can the initialization of a link-cut tree be done in time $O(n)$?