splitting mechanism with one hot encoded variables (tree based/boosting)
i have read online that if we have a categorical variable with q levels, the tree has to choose from ((2^q/2)-1) splits. For a dummy variable, there is only one possible split and this induces sparsity
i am not sure i understand this, say i have a column called color: red, green, blue,yellow , and i implement one hot encoding so is the number of splits that happen is 2^4/2 -1 = 3?...

You have the order of the operations wrong there (probably because the &quot;((2^q/2)-1)&quot; you quote above is misleading, but compare to the ESL quote in your image):  it's
$$ \frac{2^4}{2}-1 = 2^{4-1}-1 = 7$$
possible splits, namely:
(red green blue) vs (yellow)
(red green yellow) vs (blue)
(red blue yellow) vs (green)
(green blue yellow) vs (red)
(red green) vs (blue yellow)
(red blue) vs (green yellow)
(red yellow) vs (green blue)

if this increases as i have e.g. 2^25/2 -1, more splits means the tree is more likely to find a 'good split' for data at hand and lead to overfitting?...

It is certainly true there are more possible splits, and that increases the model's capacity and hence perhaps increases overfitting.  This is particularly worrisome if some of the levels are quite rare, and less so if you have a lot of data.

But what i don't understand is how this splitting chages with dummy variables.. does that equation hold or not for one hot endoded variables.

No, when you one-hot encode such a feature, the tree now must split on only one of those dummy variables (at a time).  So, considering all the $q$ levels' new indicator variables, you have exactly $q$ splits to consider.  In your example, they are the first four splits given above:
is_yellow=0 vs is_yellow=1
is_blue=0 vs is_blue=1
is_green=0 vs is_green=1
is_red=0 vs is_red=1