Given a connected, undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together. A single graph can have many different spanning trees. We can also assign a weight to each edge, which is a number representing how unfavorable it is, and use this to assign a weight to a spanning tree by computing the sum of the weights of the edges in that spanning tree.
The definition of a tree accepted by science and the forestry industry
is “A woody plant (arboreal perennial) usually with a single columnar
stem capable of reaching six metres in height”. Less than six metres
(21ft) of potential height is regarded as a shrub.This definition is not absolute; gardeners
contest the height threshold, some
preferring five metres (17ft) and others
choosing a threshold of three metres
(13ft). It is likely that bonsai enthusiasts
would entirely dismiss any figures
The uniform spanning forest (USF) in Zd is the weak limit of random, uniformly chosen, spanning trees in [−n, n]d . Pemantle  proved that the USF consists a.s. of a single tree if and only if d ≤ 4. We prove that any two components of the USF in Zd are adjacent a.s. if 5 ≤ d ≤ 8, but not if d ≥ 9. More generally, let N (x, y) be the minimum number of edges outside the USF in a path joining x and y in Zd . Then max N (x, y) : x, y ∈ Zd = (d − 1)/4 a.s.