Competition for a perch can be subtle. Look for one cat staring at the approach, occupying a connecting platform, repeatedly displacing another or following it away from the tower. The other cat may wait, turn back, use the furniture only at particular times or choose a less comfortable place. These observations are more useful than assuming that the absence of a fight means everyone is relaxed.
Watch several ordinary periods of the day and note who approaches, who leaves and whether there was a clear invitation to play. A short home video can help you explain the pattern to a professional if needed. Provide another appealing resting option with independent access while you assess the situation. Do not test the relationship by placing both cats on the same perch. If access remains restricted or either animal appears unsettled, seek advice rather than allowing repeated confrontations to continue.
Keep a simple record of who approached first, whether another cat was present and what happened next. Repeatedly losing access is a more useful pattern to discuss than labelling one isolated chase as either harmless play or definite bullying.
For product comparisons alongside this advice, see Ultimate PawHouse and Taj-MeowHall. The details below explain their sizes and uses.
Further reading: cats.org.uk; cats.org.uk; cats.org.uk.
Ultimate PawHouse
Taj-MeowHall
Cat-Cave