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What about the tree One simple design is: TREE { ID INTEGER, V INTEGER, LEFT INTEGER, RIGHT INTEGER } KEY { ID } Explanation: After it s built, the tree will contain n nodes i.e., there will be n tuples in TREE and attribute ID ( node identifier ) will take exactly the values 1, 2, ..., n. For a given node, LEFT and RIGHT refer to the left and right child, respectively; I adopt the convention that a LEFT or RIGHT value of zero means no corresponding child exists. Note: As I ve explained elsewhere (see, e.g., my book An Introduction to Database Systems, 8th edition, Addison-Wesley, 2004), INLIST, OUTLIST, and TREE are strictly speaking not relations as such but, rather, relation variables (relvars for short). From this point forward I ll be careful always to use the term relvar when it really is a relvar I m talking about, and reserve the term relation to mean a relation value (in particular, the value of some relvar).

Caution The tile index for Sprite starts with 0, but the tile index for TiledLayer starts with 1! This is a little confusing (it caused me to get an IndexOutOfBoundsException the first time I made a Sprite because I assumed that the Sprite images were numbered like the TiledLayer images). Yet the system is completely logical. In a TiledLayer, the tile index 0 indicates a blank tile (in other words, paint nothing in the cell if the cell s tile index is set to 0). A Sprite, however, comprises only one cell, so if you want that cell to be blank, then you can just call setVisible(false), meaning that Sprite doesn t need to reserve a special index to indicate a blank tile. This little confusion in the indexes shouldn t pose a big problem, but it s something to keep in mind if you can t figure out why your animation appears to be displaying the wrong images. Aside from this point, the image file is divided into individual frames or tiles in TiledLayer just as in Sprite, explained previously.

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Essentially I want to propose just one new relational operator here, which I ll call (unsurprisingly) TREESORT TREESORT takes a single relation as input and produces another as output The input relation has two attributes, which I ll assume for the moment are called ID and V and have semantics as explained in the previous subsection, and the output relation has the same heading as the input one Thus, if INLIST is as in the previous subsection, the expression TREESORT ( INLIST ) yields a relation looking like, and having the same semantics as, OUTLIST in the previous subsection Points arising: In practice we would need a way to specify which attribute is which, as it were, in the input relation, so that TREESORT knows which is the identifier attribute and which the value one I omit such details here for simplicity.

So, if(OBSTACLE) becomes if(SensorUS(SONAR)<NEAR) for the compiler, after the preprocessor has finished its work. If the Ultrasonic Sensor reading is less than NEAR (another constant with the value of 20), then the runState variable will be assigned a random value that can be 1 or -1 (corresponding to the right or left direction). Otherwise, it will be assigned the constant value 0 (WALK). The function Random(2) returns a random value that can be 0 or 1. So, runState = 1-2*(Random(2)) assigns the runState variable a value that can be 1-2*(0) = 1 or 1-2*(1) = -1.

In practice we wouldn t want TREESORT to be limited to sorting lists of integers but to work for values of any ordinal type (where an ordinal type is simply a type for which the operator < is defined) I limit my attention to integers here for simplicity It should be clear that the user of TREESORT needs to know the structure of the input and output relations but not the structure of the tree relation, which is purely internal to the TREESORT implementation Of course, we could make the structure of that tree relation externally visible (and provide user-visible BUILD and INORDER operators as well) if we had some good reason to do so.

One final point: Remarks similar to those of the present subsection apply to the discussions of the next section ( Traversing the Nodes of a Tree ) also, mutatis mutandis, and I won t bother to spell the details out again in that section..

The first step in creating your TiledLayer is to decide how many rows and columns of cells you ll need. If you don t want your layer to be rectangular, it isn t a problem because any unused cells are by default set to being blank, which prevents them from getting in the way of other images. In my example, shown in Listing 3-7 in a moment, I have only one row, and I calculate the number of columns based on the width of the screen. In the course of the game, the background tiled layer moves to the right and to the left to give the impression that the cowboy is moving (even though in reality he stays in the center of the screen), so to give plenty of room to translate the background, it is calculated to be a little more than three times the width of the screen. You can see that in the constructor Grass() in

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