Consider a minimax tree for an adversarial search problem. For example, in this picture (alpha-beta pruning):
When marking the the tree with $[\min,\max]$ values bottom-up, we first traverse node $3$ and assign $B.\max = 3$. Then we traverse $12$ and $8$ in this order, it will make sure $B.max = 3$.
But why is $B.\min = 3$? What is the use of that value?
This figure (which is in fact correct) is used in the explanation of the alpha-beta pruning algorithm on a minimax tree. Alpha-beta pruning is a method used to prune parts of the minimax tree in an adversarial search problem. In the context of a tic-tac-toe game, minimax trees are meant to allow the computer to search through the space of all possible game boards (configurations of x's and o's) assuming the player moves are optimal. This allows the computer to come up with a move that provides the best outcome (this is why the connect-four game on your computer is so incredibly difficult to beat!). For a more complete description, I highly suggest "AI a Modern Approach" by Stuart and Norvig (pg. 162-170 ish in the 2nd ed.).
Now that we've cleared up some confusion on to the algorithm. Alpha-beta pruning tries to avoid expanding subtrees based on how the minimax algorithm works. We know that the max node at the top level will take the largest value of all its children. So, node $B$ finds the value $3$, and so far, this is the maximum value its willing to pass up to its parent so it puts this value in the MAX slot. Then it finds $12$. Remember that $B$ is a MIN node, so it wants to minimize the value it passes up to its parent, thus it keeps the value $3$ in the MAX slot. Again for $8$. When $B$ has searched all of its children, it knows the maximum lower bound ($\alpha$) solution and the minimum upper bound ($\beta$) solution of its subtree and maintains those values in MIN ($\alpha$) and MAX ($\beta$) (as [3, 3]).
Note: min and max labeled in the figure are NOT the minimum and maximum values of the subtree! They are (quite confusingly labeled) the alpha-beta bounds of the solutions of the subtree (remember this is an adversarial search problem).
Next we move to node $C$. Here we come across a $2$ in the first position. Node $C$, wanting to select the lowest value from its subtree now KNOWS that its parent will not pick its value since node $B$ found a larger value already. Therefore, we can prune the rest of the subtree and continue on to $D$.
Finally, to answer the specific question: Why is $B$.min = 3? A value for $\alpha$ (the maximum lower bound of solutions at this node) and $\beta$ (the minimum upper bound of solutions at this node) is maintained at each node in order to perform the pruning. These values bound the possible cases in which the value from a node (or its subtree) may be part of the solution.
In this example, it does not appear to play a role, however, try to look at more complicated examples (i.e. trees with a height > 3) like this one and see if you can make sense of it.
I cannot do justice to minimax or alpha-beta pruning here (mostly because I haven't used them in years), so if you would really like to understand this please check out a book on AI like the one by Stuart and Norvig (the wikipedia page surprisingly has no visualization either).
