A tree graph is uniquely identified by the set of its edges
{1>2, 1>3, 2>4, 2>5, 2>6, 3>7, 3>8}
This syntax is used in Wolfram Language's graph specification. The actual picture of the graph can be displayed in Mathematica with:
Graph[{1 > 2, 1 > 3, 2 > 4, 2 > 5, 2 > 6, 3 > 7, 3 > 8}]
The question is: is there a better way to represent the graph in a compact, readable, and textonly form?
Why?
A few use cases which might give some motivation:

In tweets, sometimes, one might want to include graphs accurately represented by a short sequence of regular characters, which ideally should also be easy to parse asis (instead of only after being processed by software such as Mathematica's
Graph[...]
function.) 
When labeling/classifying objects, besides using tags which are a flat list of IDs of form
{tag1, tag2, ...}
and equivalent to a graph with vertexes but no edges, one can use a tree graph represented in a succinct form to carry more information about the hierarchical classification of the object.
One way: list of tags
One obvious way is to write the list of edges such as
{1>2, 1>3,2>4, 2>5, 2>6, 3>7,
3>8}
, which uniquely identifies the graph. It's not very
compact  vertexes with multiple children is repeated  and
neither very readable  one need to do much mental processing and
memorizing in order to understand and imagine the structure of the
graph.
Another way: "graphlet"
Another way I designed is to write it into the following form, which I dubbed "graphlet representation" of (tree) graphs:
1`{2`{4, 5, 6}, 3`{7, 8}}
So, in graphlet representation:

The edges in a graph are represented by an edge character (backtick for instance);

Child vertexes are grouped by a pair of grouping characters (
{
and}
for instance).
Note that the graphlet representation is shorter than the flat listofedges representation.
Classification graphlet is more informative than list of tags
An object's classification is often represented as a list of tags
{tag1, tag2, ..., tag8}
However, if the classification is hierarchical, a graphlet representation can easily records more information about the classification structure:
tag1`{tag2`{tag4, tag5, tag6}, tag3`{tag7, tag8}}
Application examples
To represent the classification of a problem, a sequence of key words is often used, e.g.
astrophysics, cosmology, generalrelativity, star, galaxy
One can actually additionally include its hierarchical classification structure by using a graphlet:
science`{physics`{astrophysics, cosmology, generalrelativity}, astronomy`{star, galaxy}}
Advantages
Preserving the full graph structure, many graph characteristics can be exploited, and graphtheoretic methods can be used for analyzing metadata in this form. For example, graphlets can be systematically shorten/simplified by pruning leaves.