Show that T is a maximum spanning tree for G

Question

Let G=(V,E) a connected graph, and p:E−>R a weight function. We denote |v|=n and E={e₁,e₂,...,e_m}, we suppose that ∀i p(e_i)≤p(e_i+1).

1-Show that in a connected graph G, if T and T′ are two distinct partial graphs of G then ∃e∈T such as T′+{e} creates a unique cycle.

Given T_i the family defined by:

T_{i =} $\begin{cases} E & \text{ if } i=0 \\ T_{i-1}/e_{i}& \text{ if } {i>0} \text{ and G(V,} T_{i-1}/e_{i}) \text{ is connected} \\ T_{i-1 }& \text{ otherwise} \end{cases}$

2-Show that T_m is a maximum spaninnig tree on G.
3- Give the complexity of the determination of T_m. 

 

Can some one help me in the second and third question? Thank you.