Evaluate clustering by using decision tree unsupervised learning
That's a good question. 

Is there such a thing? Can we use the tree to identify which combination will give us higher cluster purity? 

Clustering and simple decision tree fitting are used together in many cases such as:


First, like you mentioned quality of clustering can be measured by using decision tree leafs. I heard this calculation first time (I know some other measures very similar to it) but it makes sence since it still measures how are clusters are distinct from each other and dense.
Second (and most used one), fitting a decision tree by assigning cluster labels as class labels. The fit in here should be overfit (training error should be close nearly 0). This let you when you have a new customer (let's say segmentation in e-commerce) you don't have to calculate all distances and find clusters, you just predict the new customer with the tree and assign cluster = segment label to her/him. 


Also if we can do that can we just use a simple tree without even evaluate how good or bad tree is? 

No, you cannot. The tree fit well (nearly overfit). Since you want to obtain cluster separation rules in your data rather than obtain a model.

And finally, as every single method is giving us different answers how can we decide and pick which one to use to pick the right combinations?

In that stage you should think why are we doing this segmentation? With all possible cluster models try to simulate your business approach and compare results. 

Hope it helps!