Evaluate clustering by using decision tree unsupervised learning I am trying to evaluate some clustering results that a company did for some data but they used an evaluation method for clustering that i have never seen before. So i would like to ask your opinion and obviously if someone is aware of this method it would be great if he/she could explain to me the whole idea.
Clusters have been made to the data set (sample of 250000 rows and 5 features out of 500000 rows) by using k-prototypes as one of the features is categorical.
All the combinations of k= 2:10 and lambda = c(0.3,0.5,0.6,1,2,4,6.693558,10) have been made and 3 methods to figure out the best combination have been use.

Elbow method (pick the number of clusters and lambda with the min WSS)

Silhouette method pick the number of clusters and lambda with the max silhouette)

Decision tree

They build a decision tree for the data and after that they calculated for every different clustering combination the following value:
(inverse leaf size weighted within cluster purity)* cluster size/ total obs
and the picked the combination which had the max value. (k=10 and lambda=4)
So my question is:
Is there such a thing? Can we use the tree to identify which combination will give us higher cluster purity?
Also if we can do that can we just use a simple tree without even evaluate how good or bad tree is?
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?
I would really appreciate if someone can help me with that.
Thanks in advance!
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!