When you have learned a Java object using Ctrl+2 you can dump its properties using:
var obj = SeS('jTree1');
var dump = obj.DoDumpWidget(true, -1, null, null, null, null, null);
To programmatically access a simple top-level property of this object is easy:
var propValue = SeSGetJavaProperty(obj,'propName');
However to get a child object's properties, you need to first get the child object and then query that for the property in question. The following code gets a property from the first child object:
var childCount = SeSGetJavaChildrenCount(obj.instance);
var child = SeSGetJavaChildAt(obj.instance, 0);
var childPropValue = SeSGetJavaProperty(child,'propName');