Hello and welcome to the community! 

If you want to use the existing inbuilt functionality, you could reference the existing online sample and documentation regarding how to add custom images to the igTree nodes prior to the text of each node. https://www.igniteui.com/help/igtree-configure-node-images

What is more, if you require to add the images at the end of each node’s text, you could use templating for this.
Samples and online documentation how to achieve this could be found at:
Setting up Node Template in the jQuery igTree – notice a downloadable code sample is included.
API reference for the related option in use - nodeContentTemplate
Additional blog article describing “How to use templates to style the different nodes of the Ignite UI Tree control” could be found here 

In general, binding click event handler could be done using something similar to following:
$('img').bind('click', function (evt) { alert(evt.target); });
http://api.jquery.com/click/
http://api.jquery.com/bind/  

Or just use the existing igTree API and handle nodeClick event, to achieve similar functionality when clicking the node, instead the actual image at the end of the node.

You can find a very good example of how to do what you're trying to achieve here: igTree node templates

Thank you for using the Infragistics forums!

You could find the default styling/css classes used at: https://www.igniteui.com/help/api/2014.2/ui.igtree

In this case in particular, you could refer to the following:

ui-state-hover - Class defining the hover state style of the node.

ui-state-active - Class defining the selected state style of the node.