Consider the MCTS example below. Each node here is a state, the connections between nodes are actions, and transitions are deterministic.

$$

Within the nodes we have two numbers: $($the number of rollouts ending in success $/$ the total number of rollouts that were experienced from that node$).$ The value of a particular node is the ratio of these two numbers $($i$.$e$.$ a probability of success$).$ In the example below, the value of the root node is $0\mathrm{.}5.$

$$

If an action from a particular node has not been experienced, we show it ending in a black dot. Since in the beginning we start with a root node that is empty, the number of rollouts at the root is merely the sum of the rollouts of all it's children.

$$

We have indexed each of the states below with a number to the top left.

Now, assume that a greedy tree policy is used to traverse the tree to select the next node to expand, and that a simulation is run from that node, resulting in success $(+1).$ Compute the values of each of the nodes $1-5$ after the tree is updated by this return being backed up$.$

Think about how the value will get propagated back up the tree.