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We know that the clock tree synthesis is performed before signal routing. What is the specific reason for that. Or we can route both at the same time?

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A clock tree will need to be balanced, so as to minimize skew between the endpoints. Having skew on the clock tree will make it difficult to meet (and also evaluate) timing for your data paths.

I suspect that clock tree synthesis is separate for this reason; it needs to supply all cells with clock loads with a reasonably uniform distance. Once that condition is met, then the tool can move on to route your data signals.

By giving priority to the clock tree, we presumably get a more balanced tree; and in turn, that makes it easier to route our data signals subject to their timing requirements.

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