Choosing the first node in a decision tree, basic example
Some corrections in the above:


While calculating entropy of a split, you should take into account the weights -  fraction of instances in each branch:


$$
entropy(A) = 0 * (2/6) + 0.81 * (4/6) = 0.54\\
entropy(B) = 0 * (3/6) + 0.91 * (3/6) = 0.45\\
entropy(C) = 0 * (2/6) + 0.81 * (4/6) = 0.54
$$


While choosing an attribute to split, you should choose the one with lowest entropy after the split. Entropy is the uncertainty, so you want to minimise it. As an example, if the split was perfect (all 0's on one side, and all 1's on the other), then entropy will be 0.


After selection of one node, the tree is built out recursively, as you mention.