ACM Midatlantic programming contest 2014, Problem C

Thoughts:

Build a binary tree corresponding rather exactly to the way the scales look.
Do this by parsing the input.

Grammar:
System = <Number> or <Variable> or <Bar>
Number = {1..9} or <Number> followed by {0..9} // i.e. a seq of digits
Variable = {A..Z} // i.e. a cap letter)
Bar = [ <System> <System> ]
Pseudocode:

Insight: 
An array of assignments of values to variables can be maintained such that every variable is either Free, a constant, or a linear expression, ax + b, in another variable x, where a and b are constants.

Then every System has a value that can also be expressed as ax+b where a and b are numbers and x is a variable.  To do this may involve updating the array of variable assignments.

Solution:
Parse input and build tree.
The tree can be traversed in depth first fashion computing the value of each bar at postfix node traversal time (3rd visit -- coming up from below right).