If you have a binary encoding of the vertices of the graph you can represent the edges with the characteristic boolean function $\chi_E$ of the edge set $E$, i.e. $\chi_E(x,y) = 1 \Leftrightarrow$ there is an edge between the vertices encoded by $x$ and $y$.
This boolean function can be represented by e.g. Ordered Binary Decision Diagrams.
It is known that this OBDD representation is not larger than classical representations (adjacency matrix or adjacency lists) and for special graph classes (e.g. cographs and interval graphs) the OBDD size is smaller then the explicit representation (Nunkesser,Woelfel: Representation of Graphs by OBDDs). So you can hope that for good structured graphs the implicit representation is better.
In the case of trees I have shown (with the tools from the above paper) that the OBDD size is generally not better than the explicit representation and you can easily construct a tree so that the OBDD size is large.
Algorithms are called symbolic or implicit if they have access to the input graph only by this functional representation and perform functional operations to solve the problem. There is a good PhD thesis from Sawitzki about this topic but it is in german (PDF).
I think this is a kind of heuristical approach because you can't guarantee that there is a good implicit representation of the graph and so a fast symbolic alghorithm. Even if there exist a good implicit representation it is very hard to find this good one.