The RadTreeNode client-side object exposes the get_checkState function which returns TreeNodeCheckState which represents the current state of the Node's checkbox.
TreeNodeCheckState can be:
0 - Unchecked
1 - Checked
2 - Indeterminate
The tri-State checkboxes are actually rendered as <span> elements with predefined CSS classes. Images, defined in these CSS classes as backgrounds to the <span> elements represent the three states.
There are three CSS classes for each of the three states of the checkboxes: rtChecked, rtUnchecked and rtIndeterminate.
To get the <span> element of a Node at the client use the get_checkBoxElement() function as you would normally do to get the real checkbox element in non tri-state checkboxes mode.
If a Node's checkbox is clicked the client-side OnClientNodeClicking and OnClientNodeClicked events fire only for the clicked Node regardless of the CheckChildNodes property of RadTreeView.