Tur\'an problems for suspension of a balanced tree

The Tur\'an number $\ex(n,H)$ is the maximum number of edges that an $n$-vertex $H$-free graph can have. The suspension $\widehat{H}$ is obtained from $H$ by adding a new vertex which is adjacent to all vertices of $H$ and a tree is balanced if the sizes of its two color classes differ at most $1$. In this paper, we obtain a sharp bound of $\ex(n,\widehat{T})$ when $n\ge 4(4k)^6$ based on the Erd\H{o}s-S\'os conjecture. We also show the bound is sharp for infinitely many $n$ and characterize all extremal graphs. In particular, if $T$ satisfies some conditions such as $T$ contains a matching covering all vertices in one color class, then the bound is sharp for all $n$. This is a new class of graphs whose decomposition family does not contain a linear forest but we still can determine its Tur\'an number.