Exercise $2\mathrm{.}6.$ Another way to generalize the two$-$thread Peterson lock is to arrange a number of two$-$thread Peterson locks in a binary tree. Suppose $n$ is a power of two. Each thread is assigned a leaf lock which it shares with one other thread. Each lock treats one thread as thread $0$ and the other as thread $1.$

In the tree$-$lock's acquire method, the thread acquires every two$-$thread Peterson lock from that thread's leaf to the root. The tree$-$lock's release method for the treelock unlocks each of the two$-$thread Peterson locks that thread has acquired, from the root back to its leaf. At any time, a thread can be delayed for a finite duration. $($In other words, threads can take naps, or even vacations, but they do not drop dead.$)$ For each of the following properties, either sketch a proof that it holds, or describe a $($possibly infinite$)$ execution where it is violated:

mutual exclusion,

freedom from deadlock,

freedom from starvation.

Is there an upper bound on the number of times the tree$-$lock can be acquired and released between the time a thread starts acquiring the tree$-$lock and when it succeeds?