Do not use a steam cleaner unless the furniture manufacturer expressly allows it for the specific materials and components. Heat and moisture can affect finishes, adhesives, fabric backing and engineered boards, even when the surface initially looks unchanged.
A steam-cleaner setting intended for household upholstery does not establish suitability for the mixed construction of a cat tree. Check the care instructions for each section and use the recommended alternative, which may involve vacuuming, wiping or removing washable covers. If you are dealing with an infestation or contamination requiring more than routine cleaning, seek appropriate veterinary or pest-control advice and explain the furniture's materials. Avoid repeatedly soaking or heating the item in an attempt to make it hygienic. A product that cannot be cleaned adequately for the situation may need replacement rather than an unsupported treatment.
Approval for one removable fabric cover is not approval for the whole assembled tree. Treat the wooden board, adhesive-backed covering and loose cushion as separate materials when checking the cleaning instructions.
For product comparisons alongside this advice, see Linden wooden cat tower and Arbor Retreat. The details below explain their sizes and uses.
Further reading: cats.org.uk.
Linden wooden cat tower
Arbor Retreat
Floral Canopy