  Given a connected graph $G$, Kemeny's constant $\mathcal{K}({G})$ measures
the average travel time for a random walk to reach a randomly selected vertex.
It is known that when an edge is added to $G$, the value of Kemeny's constant
may either decrease, increase, or stay the same. In this paper, we present a
quantitative analysis of this behaviour when the initial graph is a tree with
$n$ vertices. We prove that when an edge is added into a tree on $n$ vertices,
the maximum possible increase in Kemeny's constant is roughly $\frac{2}{3}n,$
while the maximum possible decrease is roughly $\frac{3}{16}n^2$. We also
identify the trees, and the edges to be added, that correspond to the maximum
increase and maximum decrease. Throughout, both matrix theoretic and graph
theoretic techniques are employed.
